Student Workbook 4th Edition
ACT MASTERY ®
Math
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ACT Mastery Math
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Student Workbook
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4th Edition
MasteryPrep
ACT is the registered trademark of ACT Inc. MasteryPrep has no affiliation with ACT Inc., and the ACT Mastery program is not approved or endorsed by ACT Inc. ®
®
Table of Contents Lesson 1: Linear Equations............................................................................................................... 7 Lesson 2: Systems of Equations...................................................................................................... 19 Lesson 3: Solving Equations: Word Problems................................................................................. 35 Lesson 4: Percentages.................................................................................................................... 49
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Lesson 5: Percent Change.............................................................................................................. 63 Lesson 6: Fractions......................................................................................................................... 79
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Lesson 7: Operations....................................................................................................................... 97 Lesson 8: Substitution....................................................................................................................115 Lesson 9: Averages, Median, Mode and Range............................................................................ 129 Lesson 10: Perimeter and Line Segments..................................................................................... 143 Lesson 11: Polygon Area............................................................................................................... 157 Lesson 12: Circle Area and Circumference.................................................................................... 169 Lesson 13: Volume........................................................................................................................ 183 Lesson 14: Inequalities.................................................................................................................. 203 Lesson 15: Exponents and Roots.................................................................................................. 217
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Lesson 16: Angle Properties.......................................................................................................... 235 Lesson 17: Angles and Parallel Lines............................................................................................ 253 Lesson 18: Pythagorean Theorem................................................................................................. 273 Lesson 19: Similar Triangles.......................................................................................................... 293 Lesson 20: Trig Geometry.............................................................................................................. 307 Lesson 21: Slope........................................................................................................................... 327 Lesson 22: Function Graphs: Coordinate Plane............................................................................ 345 Lesson 23: Circles and Parabolas................................................................................................. 367
Lesson 24: Factors........................................................................................................................ 383 Lesson 25: Quadratic Equations.................................................................................................... 397 Lesson 26: Probability................................................................................................................... 413 Lesson 27: Patterns and Sequences............................................................................................. 429 Lesson 28: Counting...................................................................................................................... 447
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Lesson 29: Ratios and Proportions................................................................................................ 461 Lesson 30: Number Concepts and Properties............................................................................... 477 Lesson 31: Math Strategy.............................................................................................................. 489
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Lesson 32: Math Pacing................................................................................................................ 509
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Math Glossary............................................................................................................................... 538
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Lesson 15
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Exponents and Roots
ACT Mastery Math ®
15.1 Entrance Ticket Solve the questions below.
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What does (3x2y)2 (4x3y2)2 equal? A. 12x4y6 B. 12x48y8 C. 48x10y6 D. 124x48y8 E. 144x10y6
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1.
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2. What does (4a2b3) (5a3b4) equal? F. 9a5b7 G. 9a6b12 H. 20a5b7 J. 20a6b7 K. 20a6b12
3.
3
8q9 = ?
A. 2q3 B. 2q6 C. 4q3 D. 4q6 E. 64q3
Entrance Ticket
Learning Targets
Multiplying with Exponents
Dividing with Exponents
218
Square and Cube Roots
ACT Practice
Sum It Up
Lesson 15 – Exponents and Roots
15.2 Learning Targets 1. Recognize and utilize exponents to calculate solutions to equations 2. Use exponents in conjunction with variables to simplify expressions
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through multiplication and division 3. Recognize and use common square and cube roots to solve
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problems
Self-Assessment
Circle the number that corresponds to your confidence level in your knowledge of this subject before beginning the lesson. A score of 1 means you are completely lost, and a score of 4 means you have mastered the skills. After you
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finish the lesson, return to the bottom of this page and circle your new confidence level to show your improvement.
Before Lesson
1
2
Learning Targets
4
After Lesson
1
Entrance Ticket
3
Multiplying with Exponents
2
3
Dividing with Exponents
219
4
Square and Cube Roots
ACT Practice
Sum It Up
ACT Mastery Math ÂŽ
15.3.1 Multiplying with Exponents (25)(23)
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(25+3)
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When multiplying terms with exponents that share a common base,
Entrance Ticket
Learning Targets
Multiplying with Exponents
Dividing with Exponents
220
Square and Cube Roots
ACT Practice
Sum It Up
Lesson 15 – Exponents and Roots
15.3.1 Multiplying with Exponents
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1. 2452 · 2253
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2. (xy7) · (x3y)
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3. (d2 f 8g3) · (dg2)
Entrance Ticket
Learning Targets
Multiplying with Exponents
Dividing with Exponents
221
Square and Cube Roots
ACT Practice
Sum It Up
ACT Mastery Math ÂŽ
15.3.1 Multiplying with Exponents
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(a3)3
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When an expression with exponents is inside parentheses and there is an exponent outside the parentheses, the exponents
Entrance Ticket
Learning Targets
Multiplying with Exponents
Dividing with Exponents
222
Square and Cube Roots
ACT Practice
Sum It Up
Lesson 15 – Exponents and Roots
15.3.1 Multiplying with Exponents
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1. (2452)2
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2. (xy7)4
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3. (d2 f 8g3)3
Math Tip
Process of Elimination: It is essential to eliminate wrong answers after each step of the problem-solving process. The ACT almost always includes a few answers that will be too small or too large once you have completed a few steps. If you don’t have to complete every step of the process to get the right answer choice, you will save valuable time.
Entrance Ticket
Learning Targets
Multiplying with Exponents
Dividing with Exponents
223
Square and Cube Roots
ACT Practice
Sum It Up
ACT Mastery Math ÂŽ
15.3.2 Dividing with Exponents and Negative Exponents x5 y6
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x3
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x3 y3
x7
When dividing two terms with the exponents that share a base,
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Negative exponents can also be written as
Entrance Ticket
Learning Targets
Multiplying with Exponents
Dividing with Exponents
224
Square and Cube Roots
ACT Practice
Sum It Up
Lesson 15 – Exponents and Roots
15.3.2 Dividing with Exponents and Negative Exponents
(xy7) ÷ (x3y)
3.
(d 2f 8g3) ÷ (dg2)
4.
x10
y4 y
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x4
÷
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2.
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1. 2452 ÷ 2253
5. 6x3yz2 ÷ 3xy2z
Entrance Ticket
Learning Targets
Multiplying with Exponents
Dividing with Exponents
225
Square and Cube Roots
ACT Practice
Sum It Up
ACT Mastery Math ®
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15.3.3 Square and Cube Roots
Entrance Ticket
Learning Targets
Multiplying with Exponents
Dividing with Exponents
226
Square and Cube Roots
ACT Practice
Sum It Up
Lesson 15 – Exponents and Roots
15.3.3 Square and Cube Roots 54 be factored?
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Can
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64 m3
Math Tip
Perfect Squares: The square root of any real number can be determined by using a calculator, but you will move more quickly on the test if you memorize the squares of the numbers from 1 to 12.
Entrance Ticket
Learning Targets
Multiplying with Exponents
Dividing with Exponents
227
Square and Cube Roots
ACT Practice
Sum It Up
ACT Mastery Math ®
2
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15.4.1 Set One DO YOUR FIGURING HERE.
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1. Which of the following expressions is equivalent to –5a 3(9a 2 – 3a 4) ? A. –45a 6 – 15a 12 B. –45a 6 + 15a 12 C. –45a 5 – 15a 7 D. –45a 5 + 15a 7 E. –30a
2. The expression (5xy3)(2x2y) is equivalent to:
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F. 7x2y4 G. 7x3y3 H. 10x2y3 J. 10x2y4 K. 10x3y4
3. The expression (x8)16 is equivalent to: A. x–8 B. x2 C. x8 D. x24 E. x128
Entrance Ticket
Learning Targets
Multiplying with Exponents
END OF SET ONE STOP! DO NOT GO ON TO THE NEXT PAGE UNTIL TOLD TO DO SO.
Dividing with Exponents
228
Square and Cube Roots
ACT Practice
Sum It Up
Lesson 15 – Exponents and Roots
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15.4.2 Set Two DO YOUR FIGURING HERE.
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4. What is 4x 3 · 6x 5 ? F. 10x 2 G. 10x 8 H. 10x 15 J. 24x 8 K. 24x 15
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5. The expression (7m 2n)(6mn 3) is equivalent to: A. 13m 2 n 3 B. 13m 3 n 4 C. 42mn D. 42m 2 n 3 E. 42m 3 n 4
6. The expression 3a 2b 3 · 5a 3b 2 · 4b 2 is equal to: F. 12a 5 b 7 G. 12a 6 b 12 H. 60a 5 b 7 J. 60a 6 b 12 K. 60a 6 b 7
Entrance Ticket
Learning Targets
Multiplying with Exponents
END OF SET TWO STOP! DO NOT GO ON TO THE NEXT PAGE UNTIL TOLD TO DO SO.
Dividing with Exponents
229
Square and Cube Roots
ACT Practice
Sum It Up
ACT Mastery Math ®
2
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15.4.3 Set Three DO YOUR FIGURING HERE.
x9
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7. If x ≠ 0, then 3 equals: x A. 1 B. 3 C. x 2 D. x 3 E. x 6
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8. What does 3a 3 · 6a 5 equal? F. 9a 2 G. 9a 8 H. 9a 15 J. 18a 8 K. 18a 15
9. Calculate A.
1 2
B.
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32 − 13 . 4 2 − 13
C. 9
16
D.
10 17
E.
2 3
Entrance Ticket
Learning Targets
Multiplying with Exponents
END OF SET THREE STOP! DO NOT GO ON TO THE NEXT PAGE UNTIL TOLD TO DO SO.
Dividing with Exponents
230
Square and Cube Roots
ACT Practice
Sum It Up
Lesson 15 – Exponents and Roots
2
2
15.4.4 Set Four
10. Simplify
24 –
DO YOUR FIGURING HERE.
54 .
F. – 6 G. –6
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H. 6 J. 6 6 78
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11. Given 2 11 = 2 11 , what is the value p ? 11 p 11 F. 1 G. 11 H. 11 J. 22
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K. 121
12. If x is a real number, what is
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x 27 equivalent to?
F. x –9 1
G. x 9
H. |x 9 | J.
END OF SET FOUR STOP! DO NOT GO ON TO THE NEXT PAGE UNTIL TOLD TO DO SO.
x9
K. x 24
Entrance Ticket
Learning Targets
Multiplying with Exponents
Dividing with Exponents
231
Square and Cube Roots
ACT Practice
Sum It Up
ACT Mastery Math ®
2
2
15.4.5 Set Five DO YOUR FIGURING HERE.
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13. Find y in terms of x for y – x = 8 x , where both y and x are positive real numbers. A. 81x B. 64x C. 9x D. 8x E. 3x
14. If a ≠ 0, then (a –4) 3 = ? 1 a12 1 G. a
F.
H. a 7 J. a 12
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K. a 64
a −3 3 3 15. If a and b both ≠ 0, then (a b ) is equivalent to: b A. a –9 b 3 B. a –9 b 2 C. a –9 b –2 D. b 3 E. b 2
Entrance Ticket
Learning Targets
Multiplying with Exponents
END OF SET FIVE STOP! DO NOT GO ON TO THE NEXT PAGE UNTIL TOLD TO DO SO.
Dividing with Exponents
232
Square and Cube Roots
ACT Practice
Sum It Up
Lesson 15 – Exponents and Roots
Sum It Up Exponents and Roots Exponent The power to which a number or term is raised Ex: 3 in the expression x3
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Base The number or variable at the “bottom” of an exponent, which is multiplied by itself the number of times indicated by the exponent Ex: x in the expression x3
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Coefficient The number before the base and exponent Ex: 7 in the expression 7x3
Squared A term that is raised to the second power Ex: “five squared” is written 52 and “x squared” is written x2
Cubed A term that is raised to the third power Ex: “five cubed” is written 53 and “x cubed” is written x3
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Negative Exponent An expression with an exponent that is negative Ex: 5–3 and x–3 Rules for Operations with Exponents
When multiplying terms with exponents that have the same base, add the exponents. Ex: 3x3 · 2x4 = 6x3+4 = 6x7
When raising an expression with exponents to another power, multiply the exponents. Ex: (y3)5 = y 3 · 5 = y15
When dividing terms with exponents that have the same base, subtract the exponents. a9 Ex: 4 = a 9 – 4 = a5 a Entrance Ticket
Learning Targets
Multiplying with Exponents
Dividing with Exponents
233
Square and Cube Roots
ACT Practice
Sum It Up