Course 3 • Chapter 1 Real Numbers 3 NAME _____________________________________________ DATE __________________ PERIOD _________ Copyright © The McGraw-Hill Companies, Inc. Permission is granted to reproduce for classroom use. Lesson 2 Reteach Powers and Exponents Example 1 Write each expression using exponents. a. 7 7 7 7 7 7 7 7 = 7 4 The number 7 is a factor 4 times. So, 7 is the base and 4 is the exponent. b. y y x y x y y x y x = y y y x x Commutative Property = (y y y) · (x · x) Associative Property = y 3 x 2 Definition of exponents To evaluate a power, perform the repeated multiplication to find the product. Example 2 Evaluate (-6) 4 . (-6) 4 = (-6) (-6) (-6) (-6) Write the power as a product. = 1,296 Multiply. The order of operations states that exponents are evaluated before multiplication, division, addition, and subtraction. Example 3 Evaluate m 2 + (n - m) 3 if m = -3 and n = 2. m 2 + (n - m) 3 = (-3) 2 + (2 - (-3)) 3 Replace m with -3 and n with 2. = (-3) 2 + (5) 3 Perform operations inside parentheses. = (-3 -3) + (5 5 5) Write the powers as products. = 9 + 125 or 134 Add. Exercises Write each expression using exponents. 1. 8 8 8 8 8 2. a a a a a a 3. 5 5 9 9 5 9 5 5 Evaluate each expression. 4. 2 4 5. (-3) 5 6. ( 3 − 4 ) 3 ALGEBRA Evaluate each expression if a = 5 and b = -4. 7. a 2 + b 2 8. (a + b) 2 9. a + b 2 The product of repeated factors can be expressed as a power. A power consists of a base and an exponent. The exponent tells how many times the base is used as a factor.
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Course 3 • Chapter 1 Real Numbers 3
NAME _____________________________________________ DATE __________________ PERIOD _________
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.Lesson 2 ReteachPowers and Exponents
Example 1Write each expression using exponents. a. 7 � 7 � 7 � 7
7 � 7 � 7 � 7 = 74 The number 7 is a factor 4 times. So, 7 is the base and 4 is the exponent.
b. y � y � x � y � x
y � y � x � y � x = y � y � y � x � x Commutative Property
= (y � y � y) · (x · x) Associative Property
= y3 � x2 Defi nition of exponents
To evaluate a power, perform the repeated multiplication to fi nd the product.
Example 2
Evaluate (-6)4.
(-6)4 = (-6) � (-6) � (-6) � (-6) Write the power as a product.
= 1,296 Multiply.
The order of operations states that exponents are evaluated before multiplication, division, addition, and subtraction.
Example 3Evaluate m2 + (n - m)3 if m = -3 and n = 2.
m2 + (n - m)3 = (-3)2 + (2 - (-3))3 Replace m with -3 and n with 2.
= (-3)2 + (5)3 Perform operations inside parentheses.
= (-3 � -3) + (5 � 5 � 5) Write the powers as products.
= 9 + 125 or 134 Add.
ExercisesWrite each expression using exponents.
1. 8 � 8 � 8 � 8 � 8 2. a � a � a � a � a � a 3. 5 � 5 � 9 � 9 � 5 � 9 � 5 � 5
Evaluate each expression.
4. 24 5. (-3)5 6. (
3 −
4 )
3
ALGEBRA Evaluate each expression if a = 5 and b = -4.
7. a2 + b2 8. (a + b)2 9. a + b2
The product of repeated factors can be expressed as a power. A power consists of a base and an exponent. The exponent tells how many times the base is used as a factor.
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Course 3 • Chapter 1 Real Numbers 5
NAME _____________________________________________ DATE __________________ PERIOD _________
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The
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.
Simplify. Express using exponents.
1. k8 · k 2. t7 · t6 3. 2w2 · 5w2
4. 3e3 · 7e3 5. 4r4(-4r3) 6. (-3l2w3)(2lw4)
7. (-11w4)(-5w3x4) 8. (-4b6)(-b2c3) 9. (10t4v5)(3t2v5)
10. 59 −−
53 11. 3
8 −−
3 12. b
6 −−
b4
13. g15
−−
g7 14. 18v5
−−−
9v 15. 24a6
−−−
6a5
16. y6 ÷ y3 17. n19 −−
n11 18. 9521
−−−
9518
19. Simplify 55 · 63 · 810
−−−−−−−
53 · 6 · 89 .
20. BONUSES A company has set aside 107 dollars for annual employee bonuses. If the company has 104 employees and the money is divided equally among them, how much will each employee receive?
21. CAR LOANS After making a down payment, Mr. Valle will make 62 monthly payments of 63 dollars each to pay for his new car. What is the total of the monthly payments?
Lesson 3 Homework PracticeMultiply and Divide Monomials
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Course 3 • Chapter 1 Real Numbers 7
NAME _____________________________________________ DATE __________________ PERIOD _________
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.Lesson 4 Homework Practice Powers of Monomials
Simplify.
1. (6t5)2 2. (4w9)4 3. (12k6)3 4. (15m8)3
5. (4d3e5)7 6. (-4r6s15)4 7. [(72)2]2 8. [(32)2]3
9. ( 3 −
5 a6b9)2 10. (4x2)3(3x6)4 11. (0.6p5)3 12.
(
1 −
5 w5x3
)
2
GEOMETRY Express the area of each square below as a monomial.
13.
9c6d
14.
14g5h9
15. MEASUREMENT In the Metric System, you would need to have (104)2 grams to equal 1 metric ton. Simplify this measurement by multiplying the exponents, then simplify by finding the actual number of grams needed to equal 1 metric ton.
16. GAMING A video-game designer is using the expression 6n3 in a program to determine points earned, where n is the game level. Simplify the expression for the n2 level.
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Course 3 • Chapter 1 Real Numbers 11
NAME _____________________________________________ DATE __________________ PERIOD _________
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.Lesson 5 ReteachNegative Exponents
Example 1Write each expression using a positive exponent.
a. 7−3
7−3 = 1 −−
73 Defi nition of negative exponent
b. a−4
a−4 = 1 −−
a4 Defi nition of negative exponent
Example 2Evaluate each expression.
a. 5−4
5−4 = 1 −−
54 Defi nition of negative exponent
= 1 −−−
625 54 = 5 · 5 · 5 · 5
b. (−3)−5
(−3)−5 = 1 −−−−
(−3)5 Defi nition of negative exponent
= 1 −−−−
−243 (−3)5 = (−3) · (−3) · (−3) ·
(−3) · (−3)
Example 3Write 1 −−
65 as an expression using a negative exponent.
1 −−
65 = 6−5 Defi nition of negative exponent
Example 4Simplify. Express using positive exponents.
a. x−3 · x5
x−3· x5 = x(−3) + 5 Product of Powers
= x2 Add the exponents.
b. w−5 −−−
w−7
w−5 −−−
w−7 = w−5 − (−7) Quotient of Powers
= w2 Subtract the exponents.
ExercisesWrite each expression using a positive exponent.
1. a−8 2. 6−3 3. n−4
Evaluate each expression.
4. 7−2 5. 9−3 6. (−2)−5
Write each fraction as an expression using a negative exponent.
7. 1 −−
57 8. 1 −−
36 9. 1 −−
x8
Simplify. Express using positive exponents.
10. 4−2 · 4−4 11. r−3 · r5 12. h−2 −−−
h4
Any nonzero number to the zero power is 1. Any nonzero number to the negative n power is the multiplicative inverse of the number to the nth power.
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