7-4 More Multiplication Properties of Exponents Hubarth Algebra

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 = 5 8 ( 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

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

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.

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.

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.

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