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Course 3 • Chapter 1 Real Numbers 3

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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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Lesson 3 ReteachMultiply and Divide Monomials

The Product of Powers rule states that to multiply powers with the same base, add their exponents.

Example 1Simplify. Express using exponents.

a. 23 ! 22

23 ! 22 = 23 + 2 The common base is 2.

= 25 Add the exponents.

b. 2s6(7s7)

2s6(7s7) = (2 ! 7)(s6 ! s7) Commutative and Associative Properties

= 14(s6 + 7) The common base is s.

= 14s13 Add the exponents.

The Quotient of Powers rule states that to divide powers with the same base, subtract their exponents.

Example 2Simplify k

8 −−

k . Express using exponents.

k8 −−

k1 = k8 - 1 The common base is k.

= k7 Subtract the exponents.

Example 3

Simplify (- 2)10 ! 56 ! 63

−−−−−−−− (- 2)6 ! 53 ! 62

.

(- 2)10 ! 56 ! 63

−−−−−−−−− (- 2)6 ! 53 ! 62 = ( (- 2)10

−−−− (- 2)6 ) ! ( 5

6 −−

53 ) ! ( 63 −−

62 ) Group by common base.

= (–2)4 ! 53 ! 61 Subtract the exponents.

= 16 ! 125 ! 6 or 12,000 Simplify.

ExercisesSimplify. Express using exponents.

1. 52 ! 55 2. e2 ! e7 3. 2a5 ! 6a 4. 4x2(–5x6)

5. 79 −−

73 6. v14 −−

v6 7. 15w7 −−−−

5w2 8. 10m8 −−−−

2m

9. 25 ! 37 ! 43

−−−−−− 21 ! 35 ! 4

10. 415 ! (- 5)6

−−−−−− 412 ! (- 5)4 11. 6

7 ! 76 ! 85 −−−−−−

65 ! 75 ! 84 12. (- 3)6 ! 105

−−−−−−− (- 3)4 ! 103


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Lesson 4 ReteachPowers of Monomials

Example 1 Simplify (53)6.

(53)6 = 53 · 6 Power of a power

= 518 Simplify.

Example 2

Simplify (-3m2n4)3.

(-3m2n4)3 = (-3)3 · m2 · 3 · n4 · 3 Power of a product

= -27m6n12 Simplify.

ExercisesSimplify.

1. (43)5 2. (42)7 3. (92)4

4. (k4)2 5. [(63)2]2 6. [(32)2]3

7. (5q4r2)5 8. (3y2z2)6 9. (7a4b3c7)2

10. (-4d3e5)2 11. (-5g4h9)7 12. (0.2k8)2

Power of a Power: To fi nd the power of a power, multiply the exponents.

Power of a Product: To fi nd the power of a product, fi nd the power of each factor and multiply.


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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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