Skill Check Lesson Presentation Lesson Quiz

Skill Check 1. 6 + 7 = ? 24 2. 14 – 6 = ? 52 3. 6 • 5 = ? 15 Write the fraction in simplest form. 4. 5.

a 4 • a 3 = (a • a • a • a) • (a • a • a) = a 4 + 3 = a 7 Notice what happens when you multiply two powers with the same base. 4 factors 3 factors a 4 • a 3 = (a • a • a • a) • (a • a • a) = a 4 + 3 = a 7 7 factors This example suggests a rule for multiplying powers with the same base.

Product of Powers Property Words To multiply powers with the same base, add their exponents. Algebra a m • a n = a m + n Numbers 4 3 • 4 2 = 4 3 + 2 = 4 5

EXAMPLE 1 Using the Product of Powers Property Lake Powell Lake Powell, the reservoir behind the Glen Canyon Dam in Arizona, can hold about 10 12 cubic feet of water when full. There are about 10 27 water molecules in 1 cubic foot of water. About how many water molecules can the reservoir hold? SOLUTION Number of water molecules in reservoir = Cubic feet of water in reservoir molecules in a cubic foot • = 10 12 • 10 27 Substitute values. = 10 12 + 27 Product of powers property = 10 39 Add exponents. ANSWER Lake Powell can hold about 10 39 molecules of water.

2 x6 • x 9 = x 6 + 9 = x 15 Using the Product of Powers Property EXAMPLE 2 Using the Product of Powers Property x6 • x 9 = x 6 + 9 Product of powers property = x 15 Add exponents.

EXAMPLE 2 Using the Product of Powers Property = x 11 Add exponents. x 6 + 5 x6 • x 9 = Product of powers property 3x • 5x 5 = 3 • 5 • x 1 • x 5 Commutative property of multiplication = 3 • 5 • x 1 + 5 Product of powers property = 3 • 5 • x 6 Add exponents. = 15x 6 Multiply.

a • a • a • a • a a • a = a 5 = a 2 a • a • a • a • a a • a = Quotients of Powers There is a related rule you can use for dividing powers with the same base. The following example suggests this rule. 5 factors a • a • a • a • a a • a 1 = a 5 = a 2 a • a • a • a • a a • a = a • a • a = a 5 – 2 = a 3 2 factors 3 factors

Quotient of Powers Property Words To divide powers with the same base, subtract the exponent of the denominator from the exponent of the numerator. a m = a m – n, where a = 0 a n Algebra = 6 8 – 5 = 6 3 6 8 6 5 Numbers

3 (0.7)6 – 2 = (0.7) 4 Using the Quotient of Powers Property EXAMPLE 3 Using the Quotient of Powers Property (0.7)6 – 2 Quotient of powers property = (0.7) 4 Subtract exponents.

3 = (0.7)6 – 2 = (0.7) 4 Using the Quotient of Powers Property EXAMPLE 3 Using the Quotient of Powers Property = (0.7) 4 Subtract exponents. = (0.7)6 – 2 Quotient of powers property Quotient of powers property Subtract exponents. Divide numerator and denominator by 2.

EXAMPLE 4 Using Both Properties of Powers Simplify . 3m 5 • m 2 6m 3 = 3m 5 • m 2 6m 3 3m 5 + 2 6m 3 Product of powers property = 3m 7 6m 3 Add exponents. = 3m 7 – 3 6 Quotient of powers property = 3m 4 6 Subtract exponents. = m 4 2 Divide numerator and denominator by 3.

Lesson Quiz Simplify. 1. 2. 3g2 • 2g4 6g6 b16 3. z6 4. 5. Challenge Find a value of x that makes 5x + 3 • 5x + 4 = 523 a true statement. Explain how you found your answer. 8; Sample answer: Using the product of powers property, the left side of the equation becomes 52x + 7, and therefore 2x + 7 must equal 23. Solving the equation 2x + 7 = 23 gives x = 8.