1. 7-4 More Multiplication Properties of Exponents
Hubarth
Algebra

2. Raising a Power to a Power
For every nonzero number a and integers m and n, (𝑎 𝑚 ) 𝑛 = 𝑎 𝑚𝑛
Example ( 5 4 ) 2 = 5 4 ∙2 = ( 𝑥 2 ) 5 = 𝑥 2 ∙5 = 𝑥 10
Raising a Product to a Power
For every nonzero number a and b and integer n, (𝑎𝑏) 𝑛 = 𝑎 𝑛 𝑏 𝑛
example (3𝑥) 4 = 3 4 𝑥 4 =81 𝑥 4

3. Ex 1 Simplifying a Power Raised to a Power
Simplify (a3)4.
Multiply exponents when raising a power to a power.
(a3)4 = a3 • 4
Simplify.
= a12

4. Ex 2 Simplifying and Expression with Powers
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.

5. Ex 3 Simplifying a Product Raised to a Power
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.

6. Ex 4 Simplifying a Product Raised to a Power
Simplify (4xy3)2(x3)–3.
(4xy3)2(x3)–3 = 42x2(y3)2 • (x3)–3
Raise the three factors to the second power.
= 42 • x2 • y6 • x–9
Multiply exponents of a power raised to a power.
= 42 • x2 • x–9 • y6
Use the Commutative Property of Multiplication.
= 42 • x–7 • y6
Add exponents of powers with the same base.
16y6 x7 =
Simplify.

7. Practice
Simplify
a. (𝑎 4 ) 7 b. (𝑎 −4 ) 7
2. Simplify each expression.
a. 𝑡 2 (𝑡 7 ) −2 b. (𝑎 4 ) 2 ∙ (𝑎 2 ) 5 c. (2𝑧) 4
d. (4𝑔 5 ) −2 e. (𝑐 2 ) 3 (3𝑐 5 ) 4 f. (2𝑚 2 𝑛) 3 (5 𝑚 3 ) 2
1 𝑎 28
𝑎 28
𝑡 2 𝑡 14 = 1 𝑡 12
𝑎 18
16 𝑧 4
1 16𝑔 10
81𝑐 26
200 𝑚 12 𝑛 3