1 Notice what happens when you multiply two powers with the same base. a 4 a 3 = ( a a a a ) ( a a a ) = 4 factors 3 factors 7 factors This example suggests a rule for multiplying powers with the same base. a =a 7a 7 Rules of Exponents 4.5 LESSON

2 Product of Powers Property To multiply powers with the same base, add their exponents. Words Algebra a m a n = a m + n Numbers = = 4 5 Rules of Exponents 4.5 LESSON

3 Using the Product of Powers Property EXAMPLE 1 Lake Powell Lake Powell, the reservoir behind the Glen Canyon Dam in Arizona, can hold about cubic feet of water when full. There are about water molecules in 1 cubic foot of water. About how many water molecules can the reservoir hold? = Product of powers property = = ANSWER Lake Powell can hold about molecules of water. Substitute values. Add exponents. SOLUTION Number of water molecules in reservoir = Cubic feet of water in reservoir Number of water molecules in a cubic foot Rules of Exponents 4.5 LESSON

4 Using the Product of Powers Property EXAMPLE 2 = 3 5 x 6 Multiply. = 15x 6 Add exponents. = 3 5 x Product of powers property 3 5 x 1 x 5 Commutative property of multiplication 3x 5x 5 = Rules of Exponents 4.5 LESSON

5 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 2 factors a a a = a 5 – 2 = a 5a 5 = a 2a 2 a a a a a a = a a a a a a = 3 factors a 3 Rules of Exponents 4.5 LESSON

6 Quotient of Powers Property To divide powers with the same base, subtract the exponent of the denominator from the exponent of the numerator. Words Algebra Numbers a ma m = a m – n, where a ≠ 0 a na n = 6 8 – 5 = Rules of Exponents 4.5 LESSON

7 Using Both Properties of Powers EXAMPLE 3 Subtract exponents. Quotient of powers property Divide numerator and denominator by 3. Simplify. 3m 5 m 2 6m 36m 3 Add exponents. Product of powers property = 3m 5 m 2 6m 36m 3 3m m 36m 3 = 3m 73m 7 6m 36m 3 = 3m 7 – 3 6 = 3m 43m 4 6 = m 4m 4 2 Rules of Exponents 4.5 LESSON

8 Easy Way to Learn the Rules of Exponents laws.html