2. Powers of Products and
Quotients (5-9)

3. Powers of Products and Quotients (5-9)
What happens when you raise a power to a power?
Rule: When you raise a power to a power [Example: (am)n ], multiply the powers together. Example: (72)3 = (72) · (72) · (72) = (7 · 7) · (7 · 7) · (7 · 7) = 76 Example: (a6)2 = a6 · a6 = (a · a · a · a · a · a) · (a · a · a · a · a · a) = a12
(am)n · amn

4. Write the expression using a single exponent.
Powers of Products and Quotients
LESSON 5-9
Additional Examples
Write the expression using a single exponent.
a. (5–2)3
b. (8–4) –5
Multiply the exponents.
= 5(–2 • 3)
= 8(–4 • –5)
Simplify the exponent.
= 5–6
= 820 = 1 56
Write with a positive
exponent.

5. Multiply exponents in (b3)–2.
Powers of Products and Quotients
LESSON 5-9
Additional Examples
Simplify b2(b3)–2.
b2(b3)–2 = b2 • b3 • (–2) 
Multiply exponents in (b3)–2.
= b2 • b–6
Simplify.
= b2 + (–6)
Add exponents when multiplying powers of the same base.
Simplify.
= b–4 1 b4 =
Write using only positive exponents.

6. Find the area of a square with side length .
Powers of Products and Quotients
LESSON 5-9
Additional Examples
Find the area of a square with side length . x 4
A = s2 s = length of a side = x 4 2
Substitute for s. x2 42 =
Use the Rule for Raising a Power of a Quotient. x2 16 =
Simplify.
The area of the square is square units. x2 16

7. Powers of Products and Quotients (5-9)
What happens when you raise a product (for example, a variable and a coefficient, like 4x) to a power?
Rule: When you raise a product to a power (Example: abm, where a and b are nonzero numbers), raise each multiplicand (a and b) to the power separately. Example: (3x)4 = 34x4 = 3 · 3 · 3 · 3 · x4 = 81x4 Example:
abn = an · bn

8. Raise each factor to the second power.
Powers of Products and Quotients
LESSON 5-9
Additional Examples
Simplify (4x3)2.
(4x3)2 = 42(x3)2
Raise each factor to the second power.
= 42x6
Multiply exponents of a power raised to a power.
= 16x6
Simplify.

9. (3z5)4 = 34 • (z5)4 Raise each factor to the fourth power.
Powers of Products and Quotients
LESSON 5-9
Additional Examples
Simplify (3z5)4.
(3z5)4 = 34 • (z5)4 Raise each factor to the fourth power.
= 34 • z5 • 4 Use the Rule for Raising a Power to a Power.
= 34 • z20 Multiply exponents.
= 81z20 Simplify.

10. Raise each factor to the fourth power.
Powers of Products and Quotients
LESSON 5-9
Additional Examples
a. Simplify (–3a)4.
(–3a)4 = (–3)4(a)4
Raise each factor to the fourth power.
= 81a4
Simplify.
b. Simplify –(3a)4.
–(3a)4 = (–1)(3a)4
Multiplication Property of –1.
= (–1)(3)4(a)4
Raise each factor to the fourth power.
= –81a4
Simplify.

11. Simplify each expression. 1. (12x2yz)2 2. (–4ab3)3 3. –(–2g2h)8
Powers of Products and Quotients
LESSON 5-9
Lesson Quiz
Simplify each expression.
1. (12x2yz)2 2. (–4ab3) –(–2g2h)8
4. 5.
144x4y2z2
–64a3b9
–256g16h8 u3 6 2
3q8r s7 5 6 u6 36
243q r
s35 40 30