Powers of Products and Quotients (5-9)

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

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.

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.

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

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

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.

(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.

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.

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)3 3. –(–2g2h)8 4. 5. 144x4y2z2 –64a3b9 –256g16h8 u3 6 2 3q8r s7 5 6 u6 36 243q r s35 40 30