Global Thinkers: Mathematics 3. Secondary (sample)

SECONDARY EDUCATION

Andalusia

Mathematics 3

What are we going to learn?

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• Work systematically and solve similar, simpler problems

• Fractions

• Operations with fractions

• Exact roots

• Radicals

• Compound interest

• Common problems

• Geometric progressions

• Surprising geometric progressions

• Polynomials

• Identities

• Second-degree equations

• Solving problems with equations

• Equivalent systems

• Types of systems by number of solutions

• Test, check, use examples…

• Organise the information

• Decimal numbers

• Fractions and decimals with the calculator

• Analytical expression of a function

• Applying linear functions. Movement problems

• Pythagorean theorem

• Algebraic applications of the Pythagorean theorem

• Axes of rotation of a shape

• Surface area of three-dimensional shapes

• Rotation. Shapes with a centre of rotation

• Axial symmetries. Shapes with axes of symmetry

• Population and sample

• Making a frequency table

• Joint interpretation of x and σ

• Position parameters: median and quartiles

• Probability in irregular experiments. Law of large numbers

• Compound proportion in arithmetic problems

• Dividing polynomials

• Methods for solving systems

• Solving problems through systems

• Studying two linear functions together

• Parabolas and quadratic functions

• Area of polygons

• Area of curved shapes

• Volume of three-dimensional shapes

• Geographic coordinates

• Composition of motions

• Mosaics, friezes and rosettes

• Using the right type of graph

• Using a calculator to get x and σ

• Statistics in the media

• Probabilities in compound experiments

Let’s find the best type of tiles for the loft floor and play with squares.

Estimate the number of grains of sand on Earth and the number of stars in the Universe.

Two problems of proportional distributions.

Calculate the perimeter and area of the Koch snowflake.

Create a formula for a water tariff that penalises high consumption.

Adjust the price of water to the number of people in each family.

Systems of equations and mixtures.

Study how the Sun behaves over the course of a year and discover new things to learn.

What flies higher, the drone or the rubber bullet? Analyse their graphs.

We work with a map of Andalucía.

Looking for polygons in three-dimensional shapes.

Design your own mosaics.

Trying to measure hapiness.

Interpret the x and σ on a graph.

Continue to roll dice and reflect on the interrupted game.

NUMBERS FOR COUNTING, NUMBERS FOR MEASURING

You are going to choose between three options for the floor of a loft apartment. You’ll analyse the different alternatives and decide on the most economical. Then you’ll do the cork board challenge to determine the number of squares and rectangles on a cork wall!

1. Look at the floor of a rectangular room, covered with square tiles on the right. The side of each tile measures 40 cm.

a) What are the dimensions of the floor?

b) Calculate the size of the incomplete tiles as precisely as you can. Express them in the form of a fraction (for example, 1/3 and 1/5).

c) What proportion of the complete tiles are red tiles? And yellow tiles?

d) What is the surface area of each colour?

2. This is how we count the squares on a cork wall formed by square tiles:

a) Some squares measure 1 × 1. Others measure 2 × 2 as you can see in the picture. There are also tiles that measure 3 × 3. How many are there of each type? How many are there in total are?

b) If we look for rectangles, how many that measure 2 × 1, 3 × 1, 3 × 2... are? How many are there in total?

What are yougoing to learn to

TAKE ACTION?

OPERATIONS WITH FRACTIONS

• Operations with fractions

NATURAL NUMBERS

FRACTIONS

• Equivalent fractions

Learning by doing videos 1-minute explanations videos

OTHER WAYS OF COUNTING

INTEGERS

• Simplifying fractions

• Comparing fractions

Key language

Recurring

Over

Simplify a fraction

DECIMAL NUMBERS

• Transforming decimals into fractions

MY MATHS DICTIONARY

• Operations with integers

FRACTIONS AND DECIMALS WITH A CALCULATOR

In context

The answer is . 781 # , seven point eight one recurring. We can say 1/3 as ‘one over three’.

25/15 is 5/3 simplified / reduced / cancelled down.

1 Natural numbers

We count all the time: the days to our birthday, our marks in an exam, etc.

We use natural numbers to count. The natural numbers are 0, 1, 2, 3, ..., 10, 11, ... 100, 101, etc. They are infinite. The set of them all is called N

We can represent natural numbers on a straight line, because they are in order.

We can also use natural numbers to indicate the position of an item in an ordered sequence.

Problem solved

1 The production costs on a farm are €692 a week. They produced an average of 2 062 eggs/day last week. They pack the eggs into cartons of two-and-a-half dozen eggs and then into boxes of 10 cartons.

a) The farmer sold the boxes of eggs for €22. What was his profit?

b) How many eggs were remaining?

2 The captain of a military unit has to organise a march. If he arranges his soldiers in rows of 8, 10 or 12, there are always 2 remaining soldiers. But if he puts them into rows of 11, he has the exact number of soldiers. How many soldiers are there in the military unit?

(Note: A military unit has between 70 and 250 soldiers.)

Let’s practise!

October is the second month of Autumn and the tenth of the year.

Egg production that week → 2 062 · 7 = 14 434 Eggs in a box → (12 · 2 + 6) · 10 = 300

No. of complete boxes → 14 434 : 300 = 48 (full) (Eggs remaining → 14 434 – 300 · 48 = 34 eggs).

In an integer division D = d · C + R is fulfilled, therefore, 14 434 = 300 · 48 + 34.

Income from selling the boxes → 48 · 22 = €1 056

Profit = income – costs → 1 056  –  692 = €364

a) The profits were €364.

b) There were 34 eggs remaining.

LCM (8, 10, 12) = 23 · 3 · 5 = 120

Common multiples of 8, 10 and 12 → 120 - 240 - 360 - 480 - 600 - …

Numbers with 2 remaining when you divide by 8, 10 and 12: 122 - 242 - 362 - 482 - 602 - …

Of these, only 122 and 242 are between 70 and 250.

And of these, only 242 is a multiple of 11. Therefore, the company has 242 soldiers.

1 A farmer buys 45 calves for €475 each. Two of them get ill and die. Six months later he sells each of the remaining calves for €1 690. His production costs were €34 690. How much profit does he make on each calf?

2 A cake shop makes 7 trays of cookies. Each tray has 65 cookies. Then they pack them in bags of 8 units and sell them for €2 a bag.

During production, they break 13 of the cookies, but sell the rest. What is their profit?

2 Other ways of counting

We use natural numbers to quantify items, or to count.

When we have a set of items, we normally number them successively (1, 2, 3, 4, ...). When we get to the last item, we stop.

There are other, less simple ways of counting using elementary arithmetic operations.

– How many boxes are there in a pile 45 boxes long, 30 wide and 40 high?

Obviously, 45 · 30 · 40 = 54 000 boxes.

– How many hours are there in nine and a half weeks?

Each week has 24 · 7 = 168 h. Half a week is 3 · 24 + 12 = 84 h.

Therefore, in nine and a half weeks there are 168 · 9 + 84 = 1 596 hours.

There are even more sophisticated ways of counting. In some of them, we need to work systematically and apply specific strategies. For example:

Problem 1

How many routes are there between A and B, using the shortest route? And from B to C? How many are there between A and C, passing through B?

There are 4 different routes between A and B.

There are 6 different routes between B and C.

There are 4 ways to get from A to B, and 6 ways to get from B to C. Therefore, there are a total of 4 · 6 = 24 ways to go from A to C passing through B.

Problem 2

Six friends organise a double-round chess tournament. How many games do they have to play in total?

The total number of games is 6 · 6 = 36.

But players do not play against themselves. This means you have to subtract 6.

Total: 36 – 6 = 30 games.

2 Other ways of counting

Problem 3

There are 7 players in a basketball team. In how many ways can the manager select 5 players for a game?

There are 7 players (a, b, c, d, e, f, g) and the manager has to choose 5 of them. It is easier to eliminate two players for now.

ab ac ad ae af ag

bc bd be bf bg

cd ce cf cg de df dg ef eg fg

As you can see in the diagram, there are 21 ways to choose 2 out of 7 (the two that won’t play). Therefore, there are 21 ways to choose the 5 who play.

Problem 4

In a championship with five players, there are prizes for the top three. In how many ways can the players win?

We use a tree diagram to analyse all the possibilities.

We see that player A is the first in 4 · 3 of the possible cases.

The same is true of the other players.

Therefore, the total number of possibilities is 5 · 4 · 3 = 60.

Problem 5

A language academy offers English, French and Chinese classes. You can study one or two of these languages, but not all three. We know that 80 students choose English, 63 French and 26 Chinese. Of these, 20 study French and English, 11 English and Chinese, and 5 French and Chinese. How many students are there in the academy?

We represent and put the data in blue: Using these data, we calculate the values like this:

E (80)

11 20 5

38 49 10

F (63)

Ch (26)

Only English: 80 – 11 – 20 = 49

Only French: 63 – 20 – 5 = 38

Only Chinese: 26 – 11 – 5 = 10

We transfer the results in red to the graph.

Finally, we add up all the numbers: 49 + 20 + 38 + 11 + 10 + 5 = 133.

There are 133 students in the academy.

Look at this other way of solving the problem:

n(E) +

n(F) + n(Ch) –

– n(F and E) – n(E and Ch) –

– n(F and Ch)

Problem 6

Take Action. How many types of squares can we draw with the vertices on the grid? And... how many are there of each type?

We can draw squares that are 1 × 1, 2 × 2, 3 × 3 and 4 × 4. That’s clear. But there are more: And more:

Let’s count the 2 × 2 squares:

There are 3 with their first vertex in the top row. An equal number have their first vertex in the second row. There are three more in the third row (but not in the 4th, or the 5th row).

In total, there are nine 2 × 2 squares.

Let’s practise!

1 How many ways can we distribute 3 different books between 6 students?

2 How many ways can we get from A to B? And from B to C? And from A to C passing through B? B C

A

3 How many ways can we distribute 6 tickets between 7 people? And 8 people?

4 There are 6 players in a ping-pong competition. How many games do they have to play? List them.

5 Five friends organise a chess competition. Each player plays every other player twice. How many games do they have to play? List them.

6 How many ways can five friends sit next to each other in a row at the cinema? List them.

7 Do Activity 6 again. This time, two of the friends are partners and sit next to each other.

8 How many ways can 5 people sit around a round table?

9 Calculate the sum of the first 50 natural numbers. Do it without adding them one by one. To do this, remember that 1 + 50 = 2 + 49 = 3 + 48 = …

10 Take Action. Look at the grid in problem 6 Count the 1 × 1, 3 × 3 and 4 × 4 squares, and the other four types above.

11 Take Action. How many types of rectangles can we draw with the vertices on the grid in problem 6?

12 In the language academy in problem 5, there were 120 students: 80 study English and 63 study French. Of these, 20 study both languages. We also know that 11 study English and Chinese, and another 7 French and Chinese. How many people study Chinese?

3 Integers

Z set

We sometimes use negative numbers when we are counting.

Example

A bank balance of –234 means that the person owes 234 to the bank.

Negative integers and natural numbers form the set of integers we call Z.

We can represent integers in order on a straight line:

–7 – 6 –5 – 4 –3 –2 –1

When we add, subtract and multiply integers the result is always an integer.

Integers

When we divide two integers, the result is usually not an integer.

Absolute value of an integer

The absolute value of a number is its magnitude without the sign: | x | | 13 | = 13 | –27 | = 27

On a graph, the absolute value of a number is its distance to zero:

|– 4| = 4 |+3| = 3

– 4 0 3

Operations with integers

Problem solved

a) 7 – |5 – 8| – (4 – 12 + 6 – 11) = 7 – |–3| – (10 – 23) = = 7 – |–3| – (–13) = = 7 – 3 + 13 = 20 – 3 = 17 b) |2 · (–9)| – [–16 – 5 · (11 – 17)] : (7 – 8) = = |–18| – [–16 – 5 · (– 6)] : (–1) = = 18 – [–16 + 30] : (–1) = 18 – [14] : (–1) = 18 + 14 = 32 c) |–4 + (–3) · 2 + 5| – 2 · |3 – |4 – 5|| = |– 4 + (–6) + 5| – 2 · |3 – 1| = = |–10 + 5| – 2 · 2 = |–5| – 4 = 5 – 4 = 1

Examples

• 7 – 5 – 11 + 15 – 17 + 3 = = 7 + 15 + 3 – 5 – 11 – 17 = = (7 + 15 + 3) – (5 + 11 + 17 ) = = 25 – 33 = –8

• 3 – 5 + 8 – (4 – 13 + 6 – 11) = = 3 – 5 + 8 – 4 + 13 – 6 + 11 = = (3 + 8 + 13 + 11) –  –(5 + 4 + 6) = 35 – 15 = 20

Problems with integers

A bathyscaphe is a vehicle for doing underwater repairs. When not in use, it is secured on an oil platform, 30 m above sea level.

On a particular day, a crane moves the bathyscaphe from the platform to the water, in the following movements:

• It lifts up the bathyscaphe, by pulling a cable for 2 minutes at 2 meters per minute.

• It moves the bathyscaphe horizontally off the platform.

• It extends the cable, lowering the bathyscaphe towards the water, for 11 minutes at 3 metres per minute. And stops to change speed.

• For the next 15 minutes, it extends one metre of cable per minute.

• The bathyscaphe does some repair work for half an hour.

How long will it take to pull the bathyscaphe to the surface, at two metres per minute?

Position relative to sea level at the point where the bathyscaphe does the repair work:

30 + 2 · (+2) + 11 · (–3) + 15 · (–1) = 30 + 4 – 33 – 15 = –14 m

Time it takes the crane to pull it to the surface:

|30 + 2 · (+2) + 11 · (–3) + 15 · (–1)| : 2 = |–14| : 2 = 14 : 2 = 7 min

It takes seven minutes to pull it to the surface when the repair work is finished.

Let’s practise!

1 Order from lowest to highest: |–4|, 19, 7, 0, –6, |–5|, – 2

2 Calculate.

a) [(1 – 4) – (5 – 3) – (–6)] · [–3 + (–7)]

b) |3 – 3 · (–7) – |5 · (–8)||

c) –3(4 – 2) – 4(3 – 8) – [4 · (–5)] · [(–3) · 11]

d) |5 + |3 – 11||

e) |30 – (–20 – 9)|

3 Calculate these expressions.

a) [(1 – 7) – (8 – 3) – 2] · (15 – 11)

b) (7 – 3) · 12 + (5 – 1) · [6 – 3 · 2]

c) (–3) – 2 · (–1) + 5 · (–2) – [2 –  4 · (–7)]

d) 17 –  4 · (–3 + 6) – 2[4 – 5(2 – 3) · (–2)]

e) |26 – (– 4) · (–3) · (–3 + 2)| – |–2 + 7| · (–4)

f) (–1) + ||1 – |1 – 1 – (–1)|| · (–1) + (–1)|

4 Look at the problem with the bathyscaphe again. Write an expression to calculate the time it takes for the crane to pull the bathyscaphe from the repair work below water to the platform.

5 Find the difference in temperature (in °C) between the melting point, M. P. (from solid to liquid), and the boiling point, B. P. (from liquid to gas), of each of the chemical elements below.

a) Chlorine. M. P.: –101; B. P.: –35

b) Phosphorus. M. P.: 44; B. P.: 280

c) Mercury. M. P.: –39; B. P.: 357

d) Bromine. M. P.: –7; B. P.: 59

6 A group of friends design a game with a die, like the one on the right:

— Each player throws the die 10 times and adds up the points they get.

— If they get the same number three times, they double the points.

— If they get the same number four or more times, they triple the points.

The table shows the results of a game of three players.

How many points did they each get?

4 Fractions

We sometimes use parts of the unit to express measurements. For example, one half, three quarters... We write these measurements as fractions:

2 1 3 4 1 000 7 Numerator Denominator

A fraction is a quotient of two integers.

This quotient can be:

• An integer , 2 6 3 3 12 4 –– == dn

The numerator is a multiple of the denominator.

• A fractional number

dn

The numerator is not a multiple of the denominator.

The set of all integers and fractional numbers is called the set of rational numbers. We use the letter Q ‘quotient’ to represent them.

We can write rational numbers as a fraction and order them on a number line. Between any two numbers on the number line there are infinite other rational numbers.

We say that the visible part of an iceberg is 1/9 of its total because we consider the entire volume of the iceberg as a unit.

Simplifying fractions

We simplify or reduce a fraction if we divide, with an exact division, the numerator and denominator by the same number (other than 1 and –1).

Let’s practise!

1 Draw a number line like this in your notebook. Put these numbers in their approximate place:

When we cannot reduce a fraction with a positive denominator any more, we say that it is irreducible.

3 Simplify these fractions.

2 What fractions do these points on the number line represent?

4 Match each of the fractions below left to the correct irreducible fraction on the right.

Equivalent fractions

Every rational number has an infinite number of equivalent fractions: 3/5 = 6/10 = 9/15 = ... There is a way to recognise when two fractions represent the same rational number:

Two fractions are equivalent when we can simplify them to the same irreducible fraction.

are equivalent because

Comparing fractions

• Two fractions with the same denominator are easy to compare by looking at their numerators.

• To compare two fractions with different denominators, we find two equivalent fractions with the same ‘common denominator’.

Problem solved

1 Are these fractions equivalent?

Check:

a) 39 9 52 12 and

and

35 15 57

2 Compare

Cross-multiplication

It is a method we can use to check if two fractions are equivalent:

b a

d c = if

· a b a d c = ··ad bc =

For example: 30 18 and 35 21 are equivalent because 18 · 35 = 630 = 21 · 30

a) We find their irreducible fractions, then check if they are

they are equivalent:

They are not

5 True or false?

a) 5 2 > – 4 7 because the first one is positive and the second one is negative.

b) 3 7 > 5 2 because the first one is greater than 1 and the second one is less than 1.

c) 3 8 > 4 7 because the first one is greater than 2 and the second one is less than 2.

d) –3 8 > – 4 7 because the first one is greater than –2 and the second one is less than –2.

6 Use simplification and cross-multiplication to check if these fractions are equivalent.

and 35 21

7 Find equivalent fractions of 126 60

a) … with a numerator of 20.

b) … with a denominator of 42.

102 36 and 221 78

8 Order these fractions from smallest to largest.

5 Operations with fractions

Adding and subtracting fractions

With the same denominator

We add (or subtract) their numerators and use the same denominator.

Multiplying and dividing fractions

Product of two fractions

The numerator is the product of the numerators and the denominator is the product of the denominators:

Combined operations with fractions

With different denominator

We transform them into equivalent fractions with the same denominator, then we add or subtract their numerators.

Quotient of two fractions

Is the product of the first fraction and the inverse of the second fraction:

We first solve the expressions in brackets and then the rest of the operations.

Remember to solve products and quotients before additions and subtractions. For example:

Let’s practise!

Calculate and simplify the results.

Fraction of an amount

To find a fraction, b a , of an amount, C, we multiply b a · C

To find 5 3 of an amount, for example of €1 200, we divide it by 5 (to get one fifth) and then, we multiply it by 3. It is, we multiply the quantity by 5 3

5 3 · €1 200 = €720

A postman delivers 3/28 of a total of 4 004 letters. How many does he deliver?

3 28 of 4 004 = 3 28 · 4 004 = 3 · 28 4 004 = 3 · 143 = 429 letters

To find a fraction, b a , of another fraction, d c , of an amount, C, we multiply · b a d c C ·

Observe

The different fractions of a whole add up 1.

For example: We give 1/3 of a cake to Ana; 1/4 of it to Marcos and the rest of it to Unai. What fraction of the cake will Unai get? 1 3 1

What fraction add up 1, in each case? a) ? , ? 2 1 4 1 and b)

? ? 1 3 2 6 and c)

? ? 11 46 and d) ,, ? ? 2 1 4 11 8 and

Three people receive an inheritance of €104 000. Alberto gets 3/8, Berta gets 5/12 and Claudia gets the rest. Claudia uses 2/5 of her portion to buy a boat. How much does she have remaining?

1 –8 3 –12 5 = 24 24 910 = 24 5 is Claudia’s portion. She spends 5 2 of her portion and has 5 3 remaining:

She has 5 3 · 24 5 · 104 000 = 8 1 · 104 000 = €13 000 remaining.

5 A cyclist has completed 5/9 of today’s 216 km stage. How many kilometres did he cycle?

6 Yesterday, Karen spent €3 900. That is 3/11 of her savings. What were her total savings before?

7 A tank has 5 250 litres of water. Braulia uses 4/15 of it and Enrique 2/5. Ruperta uses the rest. Ruperta uses 3/10 of her portion to water tomatoes and the rest to water her fruit trees. How much water does Ruperta use to water her fruit trees?

6 Decimal numbers

We use decimal numbers a lot in measurements because they can express intermediate values between two integers.

We can represent decimal numbers on a number line. Using decimal numbers, we are able to approach any number of the line as much as we want.

Following this procedure, the red point can be designated by a decimal number as close as we like (3.857...).

Writing numbers in decimal form gives us a very easy and effective way to assess them, compare them and operate with them.

Types of decimal numbers

Terminating decimals

They have a limited number of decimal places.

5.4 0.97 8 –0.0725

Recurring decimals

They have an infinite number of decimal places that repeat periodically.

– In pure recurring decimals, all the figures after the decimal point repeat:

7.81818181... =  781 #

– In mixed recurring decimals, one or more of the figures after the decimal point does not repeat:

18.35222222... = 18 35 2 !

Let’s practise!

1 What types of decimal are these? 3.52 . 28 ! . 154 # 3  = 1.7320508…

2.7 3.5222… π – 2 = 1.1415926…

2 Order these numbers from smallest to largest. 25 ! 2.5 235 ! 2.505005…

3 Write three numbers between 2.5 and . 25 ! .

Remember

On calculators, like in English, we use a full stop, not a comma, as the decimal point.

1 437.54 → {∫∫‘¢«|…∞¢}

Remember

In a number, the group of decimal places that keeps repeating is called period. We draw an arc over repeating figures to indicate the period:

. 568 # 16.14 7 !

Not terminating or recurring

They have an infinite number of figures that do not repeat regularly. They are irrational numbers:

2 = 1.4142135…; π = 3.14159265…

Transforming fractions into decimals

To write a fraction in decimal form, we divide the numerator by the denominator. The quotient can be:

• An integer, when the numerator is a multiple of the denominator.

9 72 = 8; – 15 240 = –16

• A terminating decimal, if the only prime factors of the denominator of the simplified fraction are 2 and 5 (or either of them).

8 3 = 0.375; 40 123 = 3.075; 25 42 = 1.68

Look at why this is true:

If the only factors are 2 and 5, we can always write the denominator as a power of base 10.

• A recurring decimal, if the denominator of the simplified fraction has a prime factor other than 2 or 5.

11

Example

If the quotient is not a terminating decimal, why can we be sure that it is recurring? Look at the example:

3 : 7

We can write all irreducible fractions in decimal form:

• Terminating decimal , if the only prime factors of the denominator are 2 and 5.

• Recurring decimal , if the denominator has prime factors other than 2 and 5. These are both rational numbers. Decimals with infinite decimal places that do not repeat are irrational numbers.

4 True or false?

a) 3 1  = 0.333… =  . 03 !

3 3  = 3 · 0.333… = 0.999… =  09 !

Since 3 3 = 1, then 09 !  = 1.

b) . 54 !  =  . 544 #

c) . 372 #  = 3.7272727… = 3. 727 #

d) . 03 !  +  . 06 !  = 1

The quotients and remainders repeat from here on.

5 Look at the denominator of the simplified fraction only. Do these fractions transform into terminating decimals or into recurring decimals?

a) 150 44 b) 150 42 c) 1 024 101 d) 500 1 001

6 Write a value of k that makes the fraction k 84

a) An integer.

b) A terminating decimal.

c) A recurring decimal.

6 Decimal numbers

Transforming decimals into fractions

How do we transform a decimal into a fraction?

• From terminating decimals to fractions

Writing a terminating decimal as a fraction is very easy because the denominator is a power of base 10. 2.5 = 10 25 = 2 5 ; 3.41 = 100 341 ; 0.004 = 1 000 4 = 1 250

• From pure recurring decimals to fractions

Let’s look at two examples of the process:

Period with one figure: N = . 54 ! = 5.4444…

. . N N 10 54 444 5 444 10 … …

10

= = 3 The decimal part disappears when we subtract:

Period with more than one figure: N = 6.207 & = 6.207207207…

. . N N 1 000 6 207 207207 6 207207 1 000 … …

= = 3 The decimal part disappears when we subtract:

1 000

You can check these two examples by doing the divisions on a calculator.

To write a pure recurring decimal, N, as a fraction:

• We multiply N by a power of base 10 to find another number with the same decimal portion.

• When we subtract them, the result is an integer.

• We isolate N to get the answer.

7 Write as fractions.

a) 6.2 b) 0.63 c) 1.0004

d) . 35 ! e) . 01 ! f) . 27 !

g) 023 # h) 41 041 & i) 40 028 & j) 59 ! k) 7 009 & l) 099 #

8 Notice that 0 208 0 791 0 999 1 += = && &

Check by writing each addend as a fraction and then adding the fractions together.

9 Transform these decimals into fractions and then calculate.

a) . 35 ! + . 176 # – 2.103 & b) . 13 ! : . 216 !

From mixed recurring decimals to fractions

• To transform N = 2563 # into a fraction:

N = 2.5636363… We multiply by 10 to get a pure recurring decimal.

10N = 25.636363… Now, we multiply by 100 to get another number with the same decimal part.

1 000N = 2 563.636363… When we subtract this number from the previous number, the decimal part disappears. In other words, the result is an integer.

1 000N – 10N = 2 563 – 25 → 990N = 2 538 → N = 990 2 538

• Another example: N = 007 324 & = 0.07324324324…

100N = 7.324324… We get a pure recurring decimal.

100 000N = 7 324.324324… Another pure recurring decimal, with the same decimal portion.

100 000N – 100N = 7 324 – 7 → 99 900N = 7 317 → N = 99 900 7 317

Check both cases with a calculator.

To write a mixed recurring decimal, N, as a fraction:

• We multiply N twice by powers of base 10 to get two pure recurring decimals with the same period.

• When we subtract them, the result is an integer.

• We isolate N to get the fraction.

Summary

To write a recurring decimal (pure or mixed) as a fraction, we use the given number to find two pure recurring decimals with the same period. When we subtract them, the result is an integer.

We already know that decimal numbers with infinite non-repeating decimal places are irrational numbers: we cannot write them as fractions.

10 Complete the process to write these numbers as fractions.

11 Write these decimals as fractions.

a) . 62 5 ! b) . 0001 ! c) . 5018 #

12 Which of these numbers are rational? Write them as a fraction.

a) 3.51

b) 5.202002000…

c) . 503 #

d) 0.3212121…

e) π = 3.141592…

f) . 74 331 &

#

13 Transform into fractions and check that . 548

7 Fractions and decimals with a calculator

Configuration

This school year is a good time to start working with a scientific calculator. We refer to the CASIO CLASSWIZ calculator in many activities, but you can use any other calculator with similar characteristics.

We mostly use a calculator for arithmetic calculations. In this case go to � and choose 1:Calculate

It is essential to configure the calculator to receive data (INPUT) and express results (OUTPUT) in the format that we need. If you configure both in mathematics mode, it will display fractions, roots and powers in a way that is familiar to you:

• Press the configuration key �. This selects 1:Input/Output and then, 1:I Mat/O Mat (INPUT and OUTPUT in mathematics mode). It is important to configure the calculator to OUTPUT in fractions and not in mixed numbers:

• Go to the configuration menu ( �) and use the ’ arrow to move to the next screen. Select 1:Fraction result. Then, select 2:d/c

Fractions

To enter fractions, use the key and the ”’‘“ arrows.

If you enter a fraction that is not simplified, you can press = to simplify it:

Let’s practise!

1 Enter the expressions below into a calculator. Check that when you press the = key, the fractions are simplified or you get the corresponding fractions.

On scientific calculators, most keys have two secondary functions (they appear above the key). The two functions are usually in different colours:

• SHIFT → yellow

• ALPHA → red

For example, this key:

When you press it calculates the cube root.

When you press you can write recurring decimals. We will learn more about this on the next page.

From now on, when we refer to a secondary function, we will use the key. For example, to refer to the cube root function:

Operating with fractions

To do operations with fractions, just enter the chain of operations in INPUT and press the = key.

For example:

If you make a mistake when entering data, you can use the � key to go back: you will delete what is on the left of the cursor.

If you want to go back to INPUT, press the “ arrow. You can enter additional addends or correct any errors.

Decimals

We write non-recurring decimals in the usual way.

If you press � with a number in the OUTPUT, it will transform the number from a fraction into a decimal or vice versa.

Use the keys to enter a recurring decimal.

Be careful!

If you enter a terminating decimal in which one or more figures repeat ‘many’ times, the calculator will probably interpret it as a recurring decimal:

5.43434343434343 =�

5.43434343434343

Calculators have limits on entering recurring decimals and fractions due to the excessive size of the INPUT or OUTPUT. You can explore and find these limits on your own.

2 Use a calculator to find the irreducible fraction that transform into these decimal numbers.

a) 2.354 b) . 3002 # c) . 00243 # d) . 37 01 #

e) . 0125 # f) . 209 ! g) . 01233 # h) . 11 !

3 Use a calculator to solve the following. Write the result as a fraction and decimal.

4 Use a calculator to do these operations with fractions and decimal numbers. Find the results as fractions and decimal numbers (terminating or recurring).

My visual summary

Natural numbers

The numbers we use to count objects are called natural numbers.

= {0, 1, 2, 3, …, 10, 11, …, 100, 101, …} Integers

Negative integers and natural numbers form the set of integers. Z = {…, –11, –10, …, –3, –2, –1, 0, 1, 2, 3, …, 10, 11, …}

The absolute value of the numbers is its magnitude without the sign. |x | → absolute value of x. |–3| = 3; |5| = 5

Operations

2 · (–9) – [–16 – 5 · (11 – 17)] : (7 – 8) = –18 – [–16 – 5 (– 6)] : (–1) = –18 – [–16 + 30] : (–1) = = –18 – 14 : (–1) = –18 + 14 = – 4

Fractions

To simplify a fraction divide its numerator and denominator by the same number (other than 1 and –1).

A fraction is irreducible when we can’t reduce it any more and its denominator is positive.

Two fractions are equivalent when we can simplify them to the same irreducible fraction.

We can use cross-multiplication to check if two fractions are equivalent:

Operations with subtractions

Decimal numbers

Terminating decimal

Limited number of decimal places.

5.4; 0.97; 8; –0.0725

Mixed recurring decimal

There are other decimal places before the period.

18.35222222... = 18.352 (

Pure recurring decimal

The period begins after the decimal point.

7.81818181... = 7.81 (

Irrational numbers

They have an infinite number of decimal places that do not repeat periodically.

2 = 1.4142135…; π = 3.14159265…

Transforming fractions into decimals

integer 2 8 4 =

terminating decimal

The denominator of the irreducible fraction has only 2 or 5 as prime factors.

recurring decimal

The denominator has prime factors other than 2 and

Transforming decimals into fractions

• From terminating decimals to fractions

052 =

. 25

== !

• From pure recurring decimals to fractions o Period with one figure: o Period with more than one figure:

The decimal part disappears when we subtract:

The decimal part disappears when we subtract:

• From mixed recurring decimals to fractions:

The decimal part disappears when we subtract:

Exercises and problems solved

1. Calculate the total

Antía spends 2/3 of her monthly pocket money in the first half of the month. Of the remaining money, she spends 3/5 in the second half. She saves €10. How much is her monthly pocket money?

Your turn We use 1/2 the oil in a bottle, and then 1/5 of the remaining oil. There are 3 L in the bottle now. What is its capacity?

2. Calculate the fraction

Two 600 millilitre jars contain orange juice: one contains 2/3 of juice, and the other, 4/5. We add water to both jars and fill them to the top. Then we empty them into a large jug. What fraction of the liquid in the large jug is orange juice?

3. Taps and fractions

Tap A fills a water tank in 2 hours, and tap B fills the same tank in 3 hours. The tank has a drain. When the taps are closed, the drain empties the tank in 6 hours. If both taps and the drain are open, how long will it take to fill the tank?

4. Explain it with fractions

At a dance, boys and girls get into pairs. 3/4 of the boys dance with 3/5 of the girls. What fraction of the people don’t dance?

Your turn At another party, 2/6 of the boys dance with 2/5 of the girls. What fraction of the people don’t dance?

If she spends 3 2 , she has 3 1 remaining.

We calculate 3 5 of 3 1 → · 53 1 3 5 1 =

At the end of the month she has: 1 3 2 5 1 15 15 15 15 2 15 10 3 – += = cm

If 15 2 of her monthly pocket money is €10, her monthly pocket money is: 10 2 15 2 150 ·= = €75

We calculate the amount of juice in each jar:

3 2 of 600 mL → 3 2 ∙ 600 = 400 mL of juice in the first jar.

5 4 of 600 mL → 5 4 ∙ 600 = 480 mL of juice in the second jar.

The large jar contains 1 200 mL. 400 + 480 = 880 mL is juice.

Therefore, the fraction of juice in the large jug is 1 200 80 8  =  15 11

If tap A fills the tank in 2 hours, it fills 1/2 of the tank in one hour.

Tap B fills 1/3 of the tank in one hour.

The drain empties 1/6 of the tank in one hour.

If all three are open at the same time, in 1 hour they will fill: 1/2 + 1/3 – 1/6 = 2/3 of the tank

So the time they will take is: 1 : 3 2 = 2 3 h = 1.5 h = 1 h 30 min.

In pairs of girls and boys, the number of boys and girls dancing is equal. Therefore, 3/4 of the total number of boys must be equal to 3/5 of the total number of girls. We represent this data in a graph:

of the people don’t dance.

Exercises and problems

DO YOU KNOW THE BASICS?

Ways of counting

1 How many different results can we get if we throw a coin and roll a die?

2 Iñaki has 4 pairs of shorts and 5 T-shirts. How many combinations of clothes has he got? What if he also has 3 sweatshirts?

3 In a storytelling contest there are three finalists: A, B and C. How many combinations of first and second prizes are there?

Fractions and decimals

4 Group the equivalent fractions. To do this, simplify them beforehand.

143 91 ; 400 225 ; 36 24 ; 39 26 ; 539 343 ; 165 66

5 Transform the following fractions into other equivalent fractions with the same denominator. Order them from lowest to highest.

24 11 – 4 7 8 3 – 6 1

6 Enter the following numbers into your calculator. Press =. Check that it simplifies the fractions and converts the decimal numbers into irreducible fractions.

a) 51 36 b) 117 81 c) . 203 # d) . 13 01 #

7 When you simplify a fraction with a numerator greater than the denominator, some calculators give the result as an integer followed by a fraction. For example, a c b , which is the same as a +  c b . Express the following fractions as the sum of an integer and a fraction.

a) 5 8 b) 15 8 c) 7 16 d) – 2 3 e) – 3 7

8 Express the following fractions as a terminating, pure recurring or mixed recurring decimal number.

10 Express as fractions.

a) –1.03 b) 14.3 ! c) –. 25 ! d) . 032 !

e) . 00 12 # f) –. 315 # g) . 5345 ! h) . 909 !

11 Order each series of numbers from lowest to highest.

a) 3.56; . 356 ! ; . 35 ! ; . 356 #

b) ;; ;; 23.. 3 8 234 15 32 10 21 !

Operations with fractions

12 Use mental arithmetic and simplify.

a) 2 1  +  4 1 b) 1 +  2 1 c) 2 –  4 1 d) 2 1  –  3 1

e) 3 2 of 60 f) 7 12  : 3 g) 15 8  · 5 h) 5 3  ·  3 2

13 Reduce these operations as much as possible.

a) 3 2  ·  4 3 2 1 –dn  –  1 6  ·  1 6 5 3 –dn

b) 5 :  4 2 1 + dn  – 3 :  1 2 1 4 –dn

TRAINING AND PRACTICE

14 Problem solved

Calculate the value of x so that each pair of fractions is equivalent:

a) 6 x and 4 26 b) 3 x and 17 51

a) The fractions are equivalent if their cross-multiplications are equal:

x · 4 = 26 · 6 → 4x = 156 → x = 39

b) We use cross-multiplication again:

3 · 17 = 51 · x → 51 = 51x → x = 1

15 Find the value of x.

a) x 18 42 35 = b) x 32 15 12 =

c) x 81 18 45 = d) x 1122 66 5 =

13

25 9 9 13 6 23 200 17 7 5 990 233 22

9 Which of these fractions correspond to terminating decimals and which to recurring decimals? Find the answers without doing any divisions.

57 37 23

16 Problem solved

Simplify by decomposing the numerator and the denominator into factors.

2

15 60  ·

==

=

Exercises and problems

17 Apply the method in Activity 16 to simplify the following expressions.

24 Take Action. This pie chart shows a typical family budget.

18 Is it possible that the result of each of these expressions is an integer? Check without using a calculator.

19 Use a calculator to find the irreducible fractions.

SOLVE SIMPLE PROBLEMS

20 You throw a coin four times. How many different results are possible?

21 The final result of a football match is 3 - 1. What were all the possible results during the match? How many of them included a 1-1 draw?

22 Six friends organise a paddle tennis competition. All the players play all of the other players.

a) How many matches do they have to play?

b) How many matches would they play if each player plays all the other players twice?

23 Nuria and Montse play a tennis match. The winner has to win two sets. What are all the possible scores during the match?

Copy the pie chart in your notebook, substituting the percentages for fractions. If possible, simplify them.

Look at your pie chart and answer:

a) How much does a family with a budget of €2 400/month spend on clothes? And at the supermarket?

b) A family saves €288/month. What is their monthly budget?

c) Read and write True or False: housing, bills and clothes correspond to more than half of the budget.

25 Take Action. The following data express how a family spends their monthly income:

Housing: 0.4 Supermarket: . 01 6 ! Clothing: . 01 3 !

Bills: 0.1 Savings: 0.05 Other: 0.04

a) Write them all as fractions and calculate the missing fraction for leisure.

b) They spend €330 on leisure. What is their monthly budget?

26 Students at a college complete a questionnaire. These are the results:

• 7/30 of the students do not have a mobile phone.

• 400 students have a computer and a mobile phone.

• 1/6 do not have a computer.

• 1/15 have neither a computer nor a mobile phone.

How many students completed the questionnaire?

27 Ana buys some comics. All the comics are the same price. She uses 1/5 of her money to pay for 1/3 of the total price of the comics. What fraction of her money is remaining after paying for all the comics?

28 A fruit shop sells boxes of apples at €2.50 a kilo. They sell the first box for €50. This represents 5/12 of the total number of kilos. How many kilos were in each box?

THINK A LITTLE MORE

29 We buy 10 kg of strawberries to make jam. We can’t use 1/5 of the strawberries because they are damaged. We cook the remaining strawberries with an equal weight of sugar. We lose 1/4 of the weight during cooking. How many kilos of jam do we obtain?

30 There is a cycling race in four stages. On the first day, 1/15 of the cyclists abandon the race. On the second day, 1/10 of the remaining cyclists abandon the race. On the third day, 3 cyclists have an accident and abandon the race. 123 cyclists finish the race. What fraction of the cyclists participated on the second day? And on the third day? How many cyclists participated in the race?

31 A tap fills a water tank in 9 hours. If the tap and drain are both open at the same time, it takes 36 hours to fill the tank. How long does the drain take to empty the tank if the tap is closed?

32 At a party, 2/3 of the guests are boys, 3/5 of the girls have a partner and 6 of the girls are single. How many guests were at the party?

33 I spend 1/10 of the money in my piggy bank. Then, I deposit 1/15 of the remaining money in the bank. I have €36 less than the original amount. What was the original amount?

34 Two farmers, a father and his daughter, take 2 hours to plant tomatoes in a field. The father takes 6 hours to do the same work alone. How long would it take the daughter to do it alone?

35 We have two 1/2 L jugs. One of them contains 2/5 of juice, and the other, 3/4 of juice. We fill them up with water and mix them in a 1.5 L jug.

a) What fraction of the liquid in the large jug is juice? Is it more than half?

b) To reduce the concentration, we add water to the large jug to the top. What is the fraction of juice in the large jug now?

36 Divide the numbers from 1 to 10 by 11.

a) How many different decimals do you get?

b) Is that related to the fact that we are dividing by 11?

c) Can you predict the result of 23 : 11 and 40 : 11?

YOU CAN ALSO DO THIS

37 Use trial and error to complete, in your notebook, the missing numbers in these equations.

38 Complete the gaps in your notebook with the signs +, –, · or : to make each equation true:

39 We have an irreducible fraction. We add the denominator to both the terms. Then we subtract the new fraction from the original fraction and obtain the original fraction again. What’s the fraction?

DID YOU UNDERSTAND IT? THINK

40 True or false? Explain and give examples.

a) Some decimal numbers are not rational.

b) The quotient of two terminating decimals is always a terminating decimal.

c) When you add up two pure recurring decimals, you always get a pure recurring decimal.

d) We can write all integers as fractions.

e) If two positive fractions are less than 1, their product can be greater than 1.

f) When you divide two recurring decimals, you always get a recurring decimal.

41 If we write the fraction 20/13 as a decimal number, what number is in the 50th position? Is the same number in the 100th position?

42 A fraction calculator transforms fractions. When you enter b a it returns b a b a 1

1–+

For example, if you enter 3 2 , you get 5 1 . Check. You enter 2 5 and re-enter the result. If we repeat this process 10 times, what will the final fraction be? What if we repeat it 11 times?

1 Make all the four-digit numbers you can with the digits 1 and 2. How many are there?

2 In a 3rd year class, the students are going to vote for a delegate and sub-delegate. They are going to choose between 5 students. What are all the possible combinations of votes?

3 Take Action. Find all the squares that there are in the following grid (1 × 1, 2 × 2…):

4 Calculate and simplify the result.

·:

5 Write three fractions. The first must convert into a terminating decimal; the second, into a pure recurring decimal, and the third, into a mixed recurring decimal.

6 Write three numbers between each pair, below. a) 20 3 and 25 4 b) and 27..28 !!

7 Classify in terminating or recurring decimals without doing the divisions.

11 Of the 28 students in a class, 4/7 passed all the subjects. Of these, 1/4 got a grade A in all subjects. How many got an A in all subjects? What fraction of the class failed a subject?

12 At a sports centre, 1/5 of the members are above 60 years old, and 2/3 are between 25 and 60 years old.

a) What fraction of the members are 25 or younger?

b) If there are 525 members, how many are there in each age group?

13 I filled 20 3/5 litre bottles by using 1/3 of the oil I had in a barrel. How many litres were in the barrel? How many 3/4 litre bottles can I fill with the rest?

14 Three businesswomen invest in a business. The first contributes 1/3 of the capital; the second invests 2/5, and the third puts in the rest.

a) Who invested the most money?

b) After three years, they distribute the profits, and the third receives 20 000 Euros. How much do each of the others get?

89 12 113 32 23 7 18

50

8 Do the following series of operations:

Think of a number → add 1/5 to it → subtract 7/10 → add 1/2

Why do you always get your original number?

9 Zoe spends 1/3 of her money on books and 2/5 on games. She has €36 remaining. How much did she have at first?

10 There is 1 500 L of water in a tank. We use 5/12 of the water in the morning and 500 L in the afternoon, to irrigate our fruit trees. What fraction is remaining?

15 I buy a bike and pay for it in three payments. In the first payment, I pay 3/10 of the total. In the second payment, I pay 4/5 of the remaining total. In the third payment, I pay €21. What was the price of the bike?

16 A tap fills a tank in 4 hours, and a drain empties it in 6 hours. If we turn on the tap and then three hours later open the drain, in how many hours will the tank fill up?

17 ¿True or false?

a) All fractions are rational numbers.

b) All rational numbers can be a fraction.

c) A fraction always equals a recurring number.

d) A recurring decimal number is rational.

I TAKE ACTION

Let's find the best type of tiles for the loft floor and play with squares

Choose the most economic tiles for a renovation

You are renovating a loft with the dimensions 4 m × 5.1 m and height of 3.2 m. Everything is finished, apart from the floor. You can choose between three types of tiles:

• Think about their dimensions and prices. Calculate the cost of the material in each of the three cases. Remember that you cannot reuse any fragments of remaining tiles.

• The workers charge €0.50 for installing each tile and €0.20 for cutting a tile. Which of the three options is the cheapest?

How many squares and rectangles can you find?

There are cork tiles on one of the walls, arranged in a rectangle of 8 × 5, as you see on the right.

1 How many squares can you count? Start by counting the squares that measure 1 × 1. Then count the squares that measure 2 × 2, etc. For example, there are 8 · 5 = 40 squares that measure 1 × 1 and 7 · 4 = 28 squares that measure 2 × 2.

2 Count the number of non-square rectangles with sides greater than 2 on the corkboard. Start with 4 × 3, 5 × 3, 6 × 3, and so on.

3 If you arrange each vertex in a dot pattern like the one above right, how many types of squares can you get? How many squares are there that measure more than 4 × 4?

Now, it's your turn...

• Make an estimate for the floor of a room in your house, the classroom or any other space you can think of. Find out the prices per square metre for the tiles and labour.

• What types of non-square rhombuses can you find in the dotted pattern? Which ones have sides greater than 3? Count how many of each of these types there are.

POWERS AND ROOTS 2

How many stars are there in the Universe?

What can youdo to

TAKE ACTION?

Astronomer Carl Sagan said: “There are more stars in the Universe than grains of sand on all the beaches on Earth”. You are going to make an estimation of both and decide if he was right.

DO

Let’s estimate the number of grains of sand on Bolonia beach inTarifa (Cádiz). This beautiful coastline has the following dimensions: 4 km long, 70 m wide and 30 m depth.

First, let’s estimate how many grains of sand there are in a cubic milimeter. To do this, we take a miniature spoon that we fill with sand and, with the help of a magnifying glass, we count that it has 40 grains. We use 45 times the spoon to fill a miniature ladle. And then, 27 times the ladle to fill a 4 cm3 miniature bucket.

a) How many grains of sand do you estimate there are in 1 mm3, approximately?

b) Use your result to estimate the total number of grains of sand on Bolonia beach.

EXACT ROOTS

What are yougoing to learn to

TAKE ACTION?

EXPONENTIATION

SCIENTIFIC NOTATION

Key language

5 · 1022 or 5 × 1022

0.00062 or 6.2 · 10–4

Estimated to be

Learning by doing videos

1-minute explanations videos

• Powers with positive exponents

• Powers with negative exponents

• Scientific notation

RADICALS

• Some rules for working with radicals

MY MATHS DICTIONARY

In context

Fifty thousand million million million (or fifty sextillon) is the same as 5 · 1022 (five times ten to the power of twenty two).

Zero point zero, zero, zero six two is the same as six point two times ten to the power of minus four.

The number of stars is estimated to be… (–) 3.214 � f 5 Three point two one four times ten to the power of minus/negative five.

Is the same as 1 10 2 000 8 3 3 = two to the power of three over ten to the power of three is the same as eight over a thousand.

1 Exponentiation

Powers with positive exponents

Powers with positive integer exponents (1, 2, 3…) are easy to interpret.

a1  = a a n  = a · a · … · a

n times

For example: 81 = 8, (– 6)4 = (– 6) · (– 6) · (– 6) · (– 6), ·· 22 7 2 7 2 77 3 = dn

Properties

1 a m  · a n  = a m + n

2 (a · b )n  = a n  · b n

3 (a m )n  = a m · n

4 a a n m  = a m – n

5 b a b a n n n = bl

Problem solved

Calculate:

Examples

a 3  · a 4 = (a · a · a) · (a · a · a · a) = a 3 + 4

(a · b)3 = (a · b) · (a · b) · (a · b) = = (a · a · a) · (b · b · b) = a 3  · b 3

(a 2)3 = a 2  · a 2  · a 2 = (a · a) · (a · a) · (a · a) = a 2 · 3 a a aaaa aaaaaa a 1 ··· ····· 4 66 4 –==  = a 6 – 4

bl

·· ·· b a b a b a bb b aaa b a b a 3 3 3 == =

Remember

We use the following keys to calculate powers with a calculator:

For squares: x

5 x= 25

For cubes:

2 = 8

For any power: á

3 á 4 = 81

Let’s practise!

1 Reduce to one power.

a) 43 · 44 · 4 b) (56)3 c)

d) 3 15 3 3 e) 210 · 510 f)

· 55

g) (a 6  · a 3)2 : (a2  · a 4)3 h) (62)3 · 35 · (27 : 22)

2 Use the properties of powers to calculate.

a) 23 · 54 b) (65 : 24) : 35 c) 3 2 4 3 · 6 3 c c m m

d) 28 ·  2 5 4cm e) 2 20 6 6 f) 2 20 5 6

g) (33)2 : 35 h) (25)3 · [(53)4 : 23]

Powers with zero or negative exponents

Property 4 was only valid for

3

Problem solved

1 Write each number as a power of base 10:

2

6 Simplify and find the result where possible.

4

5 Write as a power of base 10.

2 Scientific notation

What problems are there when we work with very large numbers?

• We have to count the figures to know how big they are.

• Operating with them is very difficult.

Scientific notation helps us express, interpret and operate with very large quantities.

Example

– Volume of Earth:

1 080 quintillon m3 = 1.08 · 1021 m3

Remember

The words billion, trillion, quadrillion… are ‘false friends’ in English.

Spanish English

1 billion = 1012 → 109

1 trillion = 1018 → 1012

1 quadrillion = 1024 → 1015

The English numbers increase by 1 000 each time.

– Distance to the star Alpha Centauri A: 40 680 000 million km = 4.068 · 1013 km

A number expressed in scientific notation consists of:

N = a . b c d … · 10n

Integer part

A single digit other than zero.

Decimal part

The rest of the significant figures (if any).

scientific notation

Exponent

Tells us how big or how small the number is.

Integer power of base 10 Shows the order of magnitude of the number.

Scientific notation for very small numbers

There are also calculations and measurements with very small amounts (in science, medicine, microscopic biology…).

• For example, the diameter of a particular bacterium is 0.00062 mm.

To express this quantity in scientific notation, we use the negative powers of base 10:

0.00062 → 6.2 · 0.0001 → 6.2 · 10–4

This number is expressed in scientific notation.

Let’s practise!

1 Express these quantities in scientific notation.

a) 2 800 000 b) 169 000 000

c) 7 020 000 000 d) 53 420 000 000 000

2 Express these numbers with all their digits.

a) 3.6 · 105 b) 8.253 · 108 c) 2.27 · 1011

3 Write these quantities in scientific notation.

a) 0.00016 b) 0.00000387

c) 0.00000000083 d) 0.000000000000000629

4 Write these numbers with all their digits.

a) 2.65 · 10–4 b) 8.253 · 10–6 c) 2.27 · 10–11

Operations with numbers in scientific notation

Addition and subtraction

We have to convert all the numbers we are going to add or subtract to the same power of base 10. Then we can factor out the common factor.

Example

4.73 · 107 – 7.5 · 106 = 47.3 · 106 – 7.5 · 106 = (47.3 – 7.5) · 106 = = 39.8 · 106 = 3.98 · 107

Multiplication or division

We must remember that each number is formed by two factors: the decimal part and the power of base 10.

Example

a) (4.73 · 107) · (7.5 · 105) = (4.73 · 7.5) · 107 + 5 = 35.475 · 1012 = = 3.5475 · 1013

b) . . 75 10 47310 · 5 7 – = (4.73 : 7.5) · 107 – (–5) = 0.631 · 1012 = 6.31 · 1011

Problem solved

1 Express 4 782 930 663 200 in scientific notation, rounding to the billions.

2 Calculate and express the result in scientific notation:

a) 23.5 · 10 4 + 123.7 · 10 2

b) 6 284 · 10 –5 – 329 · 10 –4

c) 0.012 · 10 3 + 172 · 10 –1

4 782 930 663 200 → Rounded: 4 783 000 000 000

4 783 000 000 000 = 4.783 · 1 000 000 000 000

Expressed in scientific notation → 4.783 · 1012

a) 23.5 · 104 + 123.7 · 102 = 23.5 · 104 + 1.237 · 104 = (23.5 + 1.237) · 104 = = 24.737 · 104 = 2.4737 · 105

b) 6 284 · 10–5 – 329 · 10–4 = 628.4 · 10–4 – 329 · 10–4 = (628.4 – 329) · 10–4 = = 299.4 · 10–4 = 2.994 · 10–2

c) 0.012 · 103 + 172 · 10–1 = 12 + 17.2  = 29.2 = 2.92 · 10

5 Express 6 274 344 825 in scientific notation, rounding to the tens of millions.

6 Calculate. Write the result in scientific notation.

a) 7.25 · 104 – 5.83 · 104

b) 7.25 · 109 + 5.83 · 109

c) 7.25 · 109 + 2.1 · 108

d) 4.73 · 107 – 7.5 · 106

e) 1.8 · 109 + 2.25 · 108

f) (2.5 · 10–7) · (8 · 10–3)

g) (3.4 · 109) : (5 · 104)

7 Calculate. Express the result in scientific notation.

a) 234 · 103 + 5 231 · 102

b) 42.81 · 10–5 – 3 450 · 10–7

c) 1.592 · 104 + 2 561

d) 0.127 · 10 – 248 · 10–3

8 In 18 grams of water (H2O) there are 6.022 · 1023 elementary molecules (Avogadro’s number).

a) How many elementary molecules are there in one gram of water?

b) What is the mass of an elementary molecule?

2 Scientific notation

Calculator for scientific notation

• You can program a calculator to work in scientific notation. But, it is better not to use this mode and stay in Normal 2 mode.

• The results should be in decimals: set to decimal OUTPUT: I Mat/O Decimal.

Entering numbers

INPUT

We use the � key to enter numbers in scientific notation.

For example:

3.456 · 1012

↓

3.456 � 12

1.03452 · 10–7 ↓

1.03452 � f 7

Problem solved

1 Operate using a calculator and interpret the results:

a) (3.214 · 10–5) · (7.2 · 1015)

b) 3.214 · 10–4 – 9.58 · 10–5

c) 3.2 · 1010 + 7.3 · 10–5 –  – 4.552 · 1010

OUTPUT

• If you enter a number with a lot of figures, the calculator automatically expresses it in scientific notation.

12345678910

1.234567891 x1010

• If you enter a number with not many figures, the number appears on the screen as a decimal, not in scientific notation.

12.3 x104

123 000

• You can sequence operations entering the numbers in any of the formats. When the result has a lot of figures, the calculator expresses it in scientific notation. If not, it expresses it in normal notation and we have to interpret it.

123000000x45000

5.535 x1012

Configuring a calculator for scientific notation

Calculators can be configured to work in scientific notation (INPUT and OUTPUT). But we have to tell it the number of significant figures we want to work with:

Enter the configuration menu by pressing � and select 3:Number format. Then, select 2:Scientific not. You will see Scientif:Select 0~9. If you press 5, for example, this means you will be working with 5 significant figures.

When you are finished working in scientific notation, do not forget to return the calculator to the normal mode:

� → 3:Number format → 3:Normal → Normal:Select 1~2

And press 2.

a) 3.214 � f 5 * 7.2 � 15 = 2.31408 · 1011

b) 3.214 � f 4 - 9.58 � f 5 = 0.0002256

Now, we have to interpret the results. In scientific notation, the result is 2.256 · 10–4.

c) 3.2 � 10 + 7.3 � f 5 – 4.552 � 10 = –1.352 · 1010

As you can see in this example, when we add or subtract numbers with very different orders of magnitude using a calculator, the smaller one does not appear in the result.

9 Solve exercises 6, 7 and 8 on the previous page with a calculator.

3 Exact roots

Square roots

25  = 5, because 52 = 25. However, –25 does not exist because (–5)2 = 25.

Two square roots

32 = 9, (–3)2 = 9

Therefore, 9 has two square roots: 3 and –3. Be careful! When we write 9 , we are referring to the positive root. So: 9 = 3.

In general, if a = bn then:

Root a n = b

Radicand

Problem solved

Calculate these roots: a)

4 356

We read this expression a n as ‘the nth root of a’.

Cube roots

They behave similarly to square roots:

1 000 8 3  =  10 2 , because 10 2 3dn

Other roots

10 000 4  = 10, because 104 = 10 000.

If a n is a rational number (integer or fraction), we say it is an exact root.

000 8 .

· 9 –

b) Because we have to find 4 356 , we first check to see if 4 356 is a perfect square.

We decompose it into prime factors: 4 356 = 22 · 32 · 112.

In other words, 4 356 = (2 · 3 · 11)2 = 662. Therefore, 4 356 = 66.

c) 1 000 = 103; 64 = 43; 64 1 000 0 4 1 3 3 = 3 3  =  4 10  =  2 5

d) 243 = 35. Therefore, 243 1 5 = 3 1 .

e) 2.7 · 1013 = 27 · 1012 = 33 · 1012 = 33 · (104)3 = (3 · 104)3

Therefore, . 27 10 13 3 = (1·) 0 3 3 3 4 = 3 · 104.

f) 1.6 · 10–9 = 16 · 10–10 = 42 · 10–10 = 42 · (10–5)2 = (4 · 10–5)2

Therefore, . 16 10 · 9 –= (· ) 10 4 52 – = 4 · 10– 5 .

Let’s practise!

1 Calculate these roots.

a) 64 6 b) 216 3 c) 14 400

d) 64 1 6 e) 216 64 3 f) 1 000 3 375 3

2

True or false?

a) (–5)2 = 25, therefore 25  = –5.

b) –5 is a square root of 25.

c) 81 has two square roots: 3 and –3.

d) 27 has two cube roots: 3 and –3.

4 Radicals

Expressions that contain root symbols are called radicals.

In 12 , the only way to solve the root is to calculate its approximate decimal value. We can only express it exactly by leaving it as it is, with the radical symbol.

Some rules to operate with radicals

Product of radicals with the same index

We can put the product of two radicals with the same index under a single radical:

32··32 6 ==

Adding and subtracting radicals

We can only add two different radicals by calculating their decimal approximations. We can only add identical radicals. For example:

25 6 44 44 –33 +=33

But this additions:

32 77 –3 +

We can only calculate them as approximations or leave them with the root symbol in the expression.

Irrational numbers

Simplifying radicals

If the factorised radicand has powers with an exponent greater than or equal to the index, we take some factors out the radical.

· 18 32 32 32 · 22 == =

Calculator for finding roots

To find roots with the calculator, we use the following keys:

• Square root

í; 15 → í15 = 3.872983346

• Cube root

; 8 3 → 8 = –2

• Root of other indexes

; 32 5 → 5 32 = 2

Find the decimal expression of these roots with the calculator: 7 , 4 3 , 16 5 .

You will notice that, in all of them, there are a lot of figures with no periodicity. In fact, the decimal expression of any non-exact root has infinite non-periodic decimal figures. Therefore, they are not rational numbers: they are called irrational, and there is an infinite number of them.

Conclusion: non-exact roots are irrational numbers. π = 3.141592654... is also an irrational number.

Let’s practise!

1 Simplify the expressions when possible.

a) 85 63 – b) 35 45 + c) 25 8 –3

d) 55 –3 e) 67 f) 67 3

g) 28 h) 749 3 3 i) 55 –3 6

2 Simplify the radical when possible.

a) 35 · 24 b) 23 · 52 3 c) 5 5 4

d) 180 e) 720 f) 375 3

My visual summary

Powers with positive exponents

Powers with zero or negative exponents

If a is a rational number other than zero and

Properties:

Scientific notation

part

0 decimal part

of base 10 showing the order of magnitude of the number

Operations:

We multiply and divide in the usual way, but when we add and subtract, we first have to rewrite the addends with the same power of base 10 to take out the common factor.

We cannot operate, we must convert the first addend

SOLUTION:

My visual summary

Roots and radicals

n

If is a rational number (integer or fraction), then it is an exact root.

Radicand

Expressions that contain root symbols are called radicals.

• Square roots: 25 5 = because 52 = 25

• Cube roots: 82 3 = because 23 = 8

• Other roots: 32 2 5 = because 25 = 32

Two square roots

32 = 9, (–3)2 = 9

Therefore, 9 has two square roots: 3 and –3.

Be careful! When we write 9 , we are referring to the positive root. So: 9 = 3.

Rules:

• Product of radicals with the same index:

• Simplifying radicals: If the factorised radicand has powers with an exponent greater than or equal to the index of the root, take some factors out of the radical.

• Adding and subtracting radicals: We can only add/subtract radicals with the same index and same radicand

Exercises and problems solved

1. Size of the COVID-19 virus

The average size of the COVID-19 virus is 67 nm (nanometres).

a) Express this in millimetres and in microns (µm).

b) Compare this to the diameter of a human hair.

Your turn When we breathe, we emit aerosols: thousands of very small particles less than 5 µm in size. These can stay in the air for several hours. They are the main cause of COVID transmission. How many viral particles are there in one aerosol particle?

2. Exoplanets and spacecrafts

In June 2021, scientists discovered the exoplanet TOI-1231 b, 90 light years from Earth.

a) How many kilometres from Earth is TOI-1231 b?

b) The speed record for a manned spacecraft is 39 900 km/h (Apollo X, 1969). How long would it take Apollo X to reach TOI-1231b?

Your turn Planet Kepler-1649c is nearly 500 light years from Earth. The Millennium Falcon from Star Wars travels at 25 000 light years a day. How long would it take it to travel from Earth to Kepler-1649c?

3. Total area of a cylinder

Find the radius and total area of a cylinder 26 cm tall and 32   cm diagonal.

Your turn The height of a rectangle measures 6 cm, and its diagonal measures 18 cm. Calculate its perimeter and its area.

a) A nanometre is one billionth of a metre. A micron is one thousandth of a millimetre.

We use scientific notation to convert units and compare them.

Look at these equations:

In millimetres: 67 nm = 67 · 10–6 mm = 6.7 · 10–5 mm

In microns: 6.7 · 10–5 mm = 6.7 · 10–5 · 103 µm = 6.7 · 10–2 µm

b) The Internet tells us the approximate diameter of a hair: 0.07 mm = 7 · 10–2 mm

–=

Therefore: . 67 10 710 110 · · 045 5 2 3 –

You can put about 1 000 viral particles in a line across the diameter of one hair.

a) A light-year is a unit of length equivalent to the distance travelled by light in a year. We express it in kilometres:

Speed of light: 300 000 km/s = 3 · 105 km/s

Seconds in a year: 365 · 24 · 60 · 60 = 3.15 · 107 seconds

Distance light travels in a year: 3 · 105 · 3.15 · 107 = 9.46 · 1012 km

1 light year = 9.46 · 1012 km

90 light years = 90 · 9.46 · 1012 km = 8.51 · 1014 km

The distance between the Earth and the exoplanet TOI-1231 b is 8.51 · 1014 km.

b) To calculate the time it would take Apollo X to get there, we divide the distance above by the speed of Apolo X, 39 900 = 3.99 · 104 ≈ 4 · 104 km/h: (8.51 · 1014 km) : (4 · 104 km/h) = 2.13 · 1010 h

One year is 365.25 · 24 = 8 766 = 8.77 · 103 hours long. (2.13 · 1010) : (8.77 · 103) = 2.43 · 106 years

Appolo X would take approximately 2 430 000 years to get there.

The diagonal is the hypotenuse of a right-angled triangle with a diameter of sides, d, and height, h. We apply the Pythagorean theorem:

Exercises and problems

DO YOU KNOW THE BASICS?

Powers: properties and operations

1 Express these powers as fractions or integers without using a calculator.

a) (–3)3 b) (–2)4 c) (–2)–3

d) –32 e) – 4–1 f) (–1)–2

g) 2 1 –3dn h) 2 1 ––2dn i) 3 4 0dn

2 Write the value of n

a) 256 = 2n b) 1 27  = 3n c) –125 = –5n d)

3 Express in the form an.

a) 73 · 53 b) (–2)5 · 35 c) 3.24 : (–4)4

d) : 2 1 3 1 33ddnn e)(–7)–2 · (–3)–2 f)

4 Reduce to one power.

a) (117 · 114) : 118 b) (a 8 : a 5)4

c) (a –2)3 · a 9 d) (a –3 · a 2)–4 : a–6

e) 125 : (–3)5 f) 8 –6 · 16 –6

Powers of base 10

5 Find the value of n.

a) 0.001 = 10n b) (10 000)2 = 10n

c) 0.0000001 = 10n d) 0.00013 = 10n

6 True or false?

a) (0.001)–3 = 109 b) (0.001)4 = 1012

c) (0.01)3 = 10–6 d) (10–2)5 = (0.1)10

7 Write as powers of base 10.

a) (0.01)–5 b) 0.001 1 4dn

c) 1 10 3 3

–dn d) 10 01 5 3 2eo

8 Write two consecutive powers of base 10 with these numbers between them.

a)13 456

c) 0.18

e) 0.02

Scientific notation

b) 1 230 022 045

d) 0.008

f) 0.000007

9 Write these numbers with all their digits.

a) 4 · 107 b) 5 · 10– 4 c) 9.73 · 108

d) 8.5 · 10– 6 e) 3.8 · 1010 f) 1.5 · 10–5

10 Write in scientific notation.

a) 13 800 000 b) 0.000005

c) 4 800 000 000 d) 0.0000173

e) 153 · 104 f) 93.8 · 10– 4

11 Complete these equations.

a) 5.25 · 107 = … 106 b) 2 · 103 = … 104

c) 4.7 · 10–3 = … 10–2 d) 234 · 104 = … 103

12 Calculate and check with a calculator.

a) (2 · 105) · (3 · 1012) b) (1.5 · 10–7) · (2 · 10–5)

c) (3.4 · 10–8) · (2 · 1017) d) (8 · 1012) : (2 · 1017)

e) (9 · 10–7) : (3 · 107) f) (4.4 · 108) : (2 · 10–5)

13 Calculate, express the result in scientific notation and check with a calculator.

a) (2.5 · 107) · (8 · 103) b) (5 · 10–3) : (8 · 105)

c) (7.4 · 1013) · (5 · 10– 6) d) (1.2 · 1011) : (2 · 10–3)

e) (2 · 104)–2 : (5 · 103) f) (5 · 10–3)–1 · (8 · 103)2

Roots

14 What is the value of the following expressions?

a) – 64 b) 81 4 c) – 1

d) 1 6 e) – 9 f) –8 3

g) 25 16 h) 8 1 3 i) –1 3

15 Use a calculator.

a) –. 16 14 4 4 b) –0.064 3 c) – 5 4 –8

TRAINING AND PRACTICE

16 Match the operations to the results. They are not in order.

a–1b3 ; a–4b2 ; 3 4 a2b–1 ; a 54 3

a) : b ab a 9 4 3 2 b) ·( ) b a a 3 12 –bl

c) (6a) –1 · (3a –2) –2 d) (a –1  · b 2)2 : (ab )2

17 Simplify. Decompose into factors. Then apply the properties of powers, as in the example.

•

18

Which of these numbers are equal to 10–3?

a) . 01 10 –2

c)

0 00001

. . 001

b) 10–5 + 102

d) 10–12 · (103)3

19 Calculate and check with a calculator.

a) 3.6 · 1012 – 4 · 1011 b) 5 · 109 + 8.1 · 1010

c) 8 · 10–8 – 5 · 10–9 d) 5.32 · 10– 4 + 8 · 10– 6

20

Calculate. Write the results with all their digits.

a) 5.3 · 1011 – 1.2 · 1012 + 7.2 · 1010

b) 4.2 · 10– 6 – 8.2 · 10–7 + 1.8 · 10–5

c) (2.25 · 1022) · (4 · 10–15) : (3 · 10–3)

d) (1.4 · 10–7)2 : (5 · 10–5)

21 Which of these numbers are in scientific notation? Express in scientific notation the numbers that are not.

a) 23.4 · 105 b) 0.7 · 104 c) 1.05 · 10–3

d) 12.4 · 10–5 e) –3 · 106 f) 0.08 · 107

22 Are these triangles right-angled? Check.

a) 13  cm; 6 cm; 7 cm

b) 14  cm; 5 cm; 11  cm

c) 17  cm; 23  cm; 40 cm

23 Simplify when possible. Look at the example.

• 6 5  – 2 5  = 4 5

a) 72 42 – b) 32 – c) 43 53 –

d) 63 2 – e) 25 3 1 5 – f) 2 2 2–

24 Knowing that ·· a ab b nn n = simplify when possible.

a) · 28 b) · 39 3 3 c) 515 ·

d) 232 · 33 e) 32 · 3 f) · 12 6 33

25 Remove all the factors you can from these radicals:

a) 32 4 b) 81 3 c) 200 3

d) 50 e) 144 4 f) 250 3

26 Calculate.

a) 50 72 10 2 – + b) 80 45 20

c) 48 375 108 + d) 5 175 28 63 – +

SOLVE SIMPLE PROBLEMS

27 Jeff Bezos’ fortune grew $3.3 · 1010 in the first 5 months of 2021. The minimum salary in Spain in 2021 was €965/month. How many minimum salaries is equivalent to what Jeff Bezos got in one month, that year? How many salaries of €18 490/year (the most common salary in Spain) are equivalent to Bezos’ salary?

28 Take Action. The average length of a bacterium is 2 µm, or 2 · 10–6 m. There are approximately 5 · 1030 bacteria in the world. If we put them all in a line, how long would the line be, in kilometres? What is this distance in light-years?

29 The cost of education in an autonomous community increased from €8.4 · 106 to €1.3 · 107 in three years. By how much did it increase? Write the answer with all the digits.

30 The Chinese Tianhe-2 computer completes 34 000 trillion operations a second. How many operations can it complete in 1 millisecond, 1 microsecond and 1 nanosecond? Calculate.

Exercises and problems

31 Target 15.2. The United Nations says that the global forest area reduced by 4.7 million hectares a year between 2010 and 2020. Calculate how many hectares of forest were lost between those years. Compare that figure with the total area of Spain.

32 Take Action. A cubic centimetre of water contains 3.35 · 1022 water molecules. There are approximately 1.39 · 109 km3 of water on Earth. How many water molecules is that? And how many molecules are there in a 2/5 litre glass?

33 We used 6.87 million tons of paper in Spain in 2019 and a little less in 2020. We recycled 4.42 million tons of paper and cardboard in 2019. Calculate the annual consumption and the amount recycled per inhabitant in Spain in 2019. (Population of Spain in 2019: 46.1 million. Source: INE).

34 In 2020, Jeff Bezos, the billionaire who traveled to space to boost the spaceflight business, made the largest charitable donation of the year, contributing $1010 to combat climate change. In that year, his fortune increased from $1.14 · 1011 to $1.79 · 1011. What percentage of his fortune did the donation represent? How much is the equivalent for a person on a €18 490 salary a year?

35 The radius of a CD is 6 cm. Can you put it in a square case with diagonal of 16 cm? And a square case with diagonal of 17 cm?

36 The Great Pacific Garbage Patch is growing very fast. This collection of marine waste contains about 80 000 tons of plastic and has a surface area of 1.6 million km2. That is nearly three times the size of France.

a) Express the data in scientific notation.

b) With these data, calculate the area of France. Find the area of France on the Internet. Compare the two figures to check the information above is true.

37 The objective of The Ocean Cleanup project is to clean up the Great Pacific Garbage Patch in 5 years. They use technology to collect and remove waste from the ocean. How much waste do they need to remove every day, to complete their goal?

38 Florence Griffith holds the world record in the women’s 100 m, with 10.49 seconds. The speed of the Apollo X rocket was 39 900 km/h. Calculate how long it takes Griffiths and Apollo X to travel one metre. Express them in scientific notation and compare them. How many times faster is Apollo X?

THINK A LITTLE MORE

39 The thermal power plants of one autonomous community in Spain emitted 5.96 million tons of CO2 into the atmosphere in 2020. For new petrol cars, the average emission is 143 g/km, and for electric cars, 68 g/km.

a) New cars travel approximately 25 000 km a year. How many tons of CO2 does a new petrol vehicle emit in a year? And an electric car?

b) How many new petrol cars are equivalent to the power plants in the autonomous community above, with reference to CO2 emissions in 2020?

40 The galaxy M87 is 50 million light-years from Earth. It has a black hole with a diameter of 60 light-years and its mass is two billion times the mass of the Sun.

a) Calculate the mass of the black hole in kilograms. (The mass of the Sun is approximately 2 · 1030 kg).

b) Express the distance from M87 to Earth in kilometres, and the diameter of the black hole.

41 The MCV (Mean Corpuscular Volume) is the average volume of red blood cells. Adolescents have an MCV of 90 femtolitres (1 fL = 10–15 L). An average adolescent has 5 L of blood and 5 million red blood cells per mm3. How many litres of red blood cells do they have?

42 Find the perimeter and area of rectangle ABCD: AB = 3 + 2  cm and BC = 1 + 2 2  cm. Write your answers as a + b 2 , where a and b are natural numbers.

43 The volume of the Moon is 2.2 · 1010 km3. How many kilometres is its radius? Use a calculator and give your answer in scientific notation.

44 The volume of a cube is 1 728 cm3. Calculate its edge, the total area, the diagonal of a face and the diagonal of the cube.

45 Find the volume of a cone with a generatrix of 18 cm and a base radius of 82  cm.

Vcone = 3 1 πr 2h

46 You have a square with sides of 1 + 3  cm and a rectangle with a height of 1 cm and a variable base:

a) What is the base of the rectangle if the two shapes have the same perimeter?

b) What is the base of the rectangle if the area of the square is double the area of the rectangle?

YOU CAN ALSO DO THIS

47

Look at these equations:

22 + 2 = 32 – 3 72 + 7 = 82 – 8 Make a hypothesis, and then prove it.

48 Factorise the number 8 000.

Calculate the value of x so that 8 0002 = x3

49 Complete in your notebook.

a) 12 34 44 = b) ·8 6 44 =

c) 6 610 4 4 += d) 71 3 –44 =

50 What is the last digit of 257? What are the last digits of the powers of base 2? Explain.

51 Find the number of figures in 28, 58 and 108. Do the same for 211, 511 and 1011. Can you make a general rule?

If the power 2456 has m figures and 5456 has n figures, how many figures does 10456 have?

DID YOU UNDERSTAND IT? THINK

52 True or false? Justify and give examples.

a) The power of a negative number can be 1.

b) If x < 0, then –x 3 > 0.

c) –x 2 is always a positive number.

d) The cube of a negative number is always less than the number.

e) If a2  = b2, then a = b

f) An integer and its square are always either both even, or both odd.

g) Half of 64 is 32 .

53 Write each of the following numbers in the form 2p · 5q, where p and q, are integers:

a) 1 000 b) 500–1 c) 80–2 d) 0.256 e) 0.05

54 Find the value of a

a) a 3 = 26 b) a –1 = 2 c) a 5 4 =

d) a 4  = 1 e) a –2  =  4 1 f) a –5 = –1

55 What is the area of a square with a perimeter of 20 3 m?

56 Order the numbers n, n2 , n and 1/n

a) If n > 1. b) If 0 < n < 1.

57 True or false?

a) ()22 –2 = b) () 2 2 1 3 3 –=

c) 16 25 9 += d) ()() 10 10 10 4 =

58 What are the GCD and the LCM of these numbers?

3 · 103 4 · 104 5 · 105 6 · 106

59 Prove that the number 3 +  2 is the inverse of 3 –  2 . Remember that its product must be equal to 1. Are the numbers 6 7 1 + and 7 –  6 equal?

60 Look at the operation below. What result would you get on your calculator? Explain your answer.

10 100 20 20 +

10

Self-assessment

GO TO THE ESCAPE ROOM AND TEST YOURSELF!

anayaeducacion.es Answer key.

1 Calculate.

(–3)–2 + 4 3 –1cm – 2–3 1 1 2 ––2cm

2 Simplify.

a) ab ab 6 3 21 2 –

–b) · a b a –1 3 2 –– c b m l

8 Take Action. There are 12 000 grains of rice in a cup. There are 20 cups in a 2.5 L bottle. Estimate how many grains of rice there are in 12 silos of 1 300 m3 volume each. Express the result in scientific notation.

c) · b a b a 2 3 4 –bl d) : () a b a b 3 4 21 ––

– cm

3 Apply the properties of powers to simplify this expression. ·( )

12 9 62 · –

4 2 3 2–

4 Write in scientific notation.

a) 758 · 10–5 b) 0.035 · 1013

c) 101 · 1011 d) 0.1001 · 10–7

5 Calculate and check your answers with a calculator.

a) (3.5 · 107) · (8 · 10–13)

b) (9.6 · 10–8) : (3.2 · 1010)

c) (2.7 · 108) + (3.3 · 107)

6 True or false?

a) The exponent of a power cannot be a negative number if the base is negative.

b) A power with a negative exponent is always negative.

c) If an = –1, then a = –1 and n is an odd number.

d) 5–2 is a negative number.

e) 6–1 < 1

f) 10–3 . 10–2 = 0.1

g) (–8)–2 = (+8)2

7 A capsule of medicine has two hemispheres that are 7 mm in diameter and a cylinder that is 14 mm long. A laboratory produces 4 tonnes of the medicine and wants to make capsules that contain 500 mg each. How many capsules can it produce? Find the volume of each capsule.

14 mm

9 R136a1 is a blue hypergiant star that has a mass equivalent to 265 solar masses and is 165 000 light years away.

Express the distance in kilometres and the mass in kilograms.

10 These are the masses of some planets:

Earth: 5.98 · 1024 kg Mars: 6.42 · 1023 kg

Jupiter: 1.90 · 1027 kg

a) How many more kilos does Earth weigh than Mars?

b) How many times heavier is Jupiter than Mars?

11 The largest natural gas reserve in Central Asia has a volume of 9 . 1011 m3. Its annual production is 1.8 . 1013 litres. How many years of energy is there remaining the reserve?

12 The radius of a circle is 2.5 cm. What is the radius of another circle with twice the area? Use a calculator and give the exact value and the decimal expression rounded to the hundredths.

mm

I TAKE ACTION

LEA RNING EXPERIENCE

Estimate the number of grains of sand on Earth and the number of stars in the Universe

Was astronomer Carl Sagan right?

Let’s count the grains of sand on the Earth’s beaches. Look at the steps in Review what you know on page 46. We got a final result of 12 grains of sand in 1 mm3 .

The length of all the beaches on Earth is around 300 000 km. Each beach is, on average, 50 m wide and 25 m deep:

• Using this information, how many grains of sand are there on the Earth’s beaches?

Count the number of stars in the Universe

Our galaxy, the Milky Way, has about 200 billion stars. It is a normal galaxy, but there are dwarf galaxies, with 107 stars, and giant galaxies, with 1014. Using photos from the Hubble Space Telescope (1990) astronomers estimated that there were approximately 2 · 1012 galaxies. However, the New Horizons spacecraft (2006) reduced this number to a few hundred billion galaxies in the visible Universe. We will have to wait and see what information the James Webb Space Telescope tells us.

• Find out how many stars there are in the Universe using the Hubble telescope estimate of the number of galaxies. Use the Milky Way as a model galaxy for the number of stars.

Think about your results and decide: Was Carl Sagan correct?

Now, it's your turn...

• Look on the Internet to find out how many square kilometres of desert there are on Earth. Investigate the average depth of sand and estimate how many grains of sand there are in the world’s deserts.

• Estimate the number of drops of water in the oceans. To obtain the volume of a drop of water, use a similar method to the one on page 46 to count the number of grains of sand.

• If you put all the grains of sand on Earth in a line, how long would it be?

James Webb Space Telescope's mirror at NASA.

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