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Example 10 Converting to index form

Simplify the following expressions by writing them in index form.

a 5 × 5 × 5 × 5 × 5 × 5

SOLUTION a 5 × 5 × 5 × 5 × 5 × 5 = 56 b 3 × 3 × 2 × 3 × 2 × 3

= 2 × 2 × 3 × 3 × 3 × 3

= 22 × 34

Now you try

b 3 × 3 × 2 × 3 × 2 × 3

EXPLANATION

The number 5 is the repeated factor and it appears six times.

First, write the factors in increasing order (all the 2s before the 3s)

The number 2 is written two times, and the number 3 is written four times.

Simplify the following expressions by writing them in index form.

Example 11 Expanding a power

Expand and evaluate the following terms.

a 24

SOLUTION b 23 × 52

EXPLANATION

Write 2 down four times and multiply. b a

Write the number 2 three times, and the number 5, two times.

Now you try

Expand and evaluate the following terms.

a 53 b 23 × 102

Example 12 Evaluating expression with powers

SOLUTION

Write in expanded form (optional step). Powers are evaluated before the subtraction occurs. b

Write in expanded form (optional step). Follow order of operation rules. Carry out the multiplication first, then carry out the addition.

Now you try

Exercise 2E

1 Simplify the following expressions by writing them in index form.

2 Simplify the following expressions by writing them in index form.

3 Simplify the following expressions by writing them as powers.

Simplify by writing using powers.

5 Expand these terms. (Do not evaluate.)

6 Expand these terms. (Do not evaluate.)

7

8

9 Determine the index number for the following basic numerals.

10 Write one of the symbols a Show, with calculations, why at least 156 people (including Jane) will have read the email within 15 minutes of Jane sending it. b If the email always goes to a new person, and assuming every person in Australia has an email address and access to email, how long would it take until everyone in Australia has read the message? (Australian population is approx. 26 million people.) c How many people will read the email within 1 hour? d Using the same assumptions as above, how long would it take until everyone in the world has read the message? (World population is approx. 8 billion people.) e Approximately how many people will have read the email in 2 hours?

, = or > in the box to make the following statements true.

11 A text message is sent to five friends. Each of the five friends then forwards it to five other friends and each of these people also sends it to five other friends. How many people does the text message reach, not including those who forwarded the message?

12 Jane writes a chain email and sends it to five friends. Assume each person who receives the email reads it within 5 minutes of the email arriving and then sends it to five other people who have not yet received it.

13 Write the correct operation (+, −, ×, ÷ ) in the box to make the following equations true.

32 42 = 52

24 42 = 44