7-3 Multiplication Properties of Exponents Hubarth Algebra

Multiplying Powers With the Same Base For every nonzero number a and integers m and n, 𝑎 𝑚 ∙ 𝑎 𝑛 = 𝑎 𝑚+𝑛 . Example 3 5 ∙ 3 4 = 3 5+4 = 3 9 (ℎ 2 ) ℎ 9 = ℎ 2+9 = ℎ 11

Ex 1 Multiplying Powers Rewrite each expression using each base only once. Add exponents of powers with the same base. 73 + 2 = a. 73 • 72 = 75 Simplify the sum of the exponents. Think of 4 + 1 – 2 as 4 + 1 + (–2) to add the exponents. 44 + 1 – 2 = b. 44 • 41 • 4–2 = 43 Simplify the sum of the exponents. Add exponents of powers with the same base. 68 + (–8) = c. 68 • 6–8 = 60 Simplify the sum of the exponents. Use the definition of zero as an exponent. = 1

Ex 2 Multiplying Powers in an Algebraic Expression Simplify each expression. a. p2 • p • p5 Add exponents of powers with the same base. p 2 + 1 + 5 = = p 8 Simplify. 2q • 3p3 • 4q4 b. Commutative and Associative Properties of Multiplication (2 • 3 • 4)(p3)(q • q 4) = Multiply the coefficients. Write q as q1. = 24(p3) (q1• q 4) Add exponents of powers with the same base. = 24(p3) (q1 + 4) Simplify. = 24p3q5

Ex 3 Multiplying Numbers in Scientific Notation Simplify (3  10–3)(7  10–5). Write the answer in scientific notation. Commutative and Associative Properties of Multiplication (3 • 7)(10–3 • 10–5) (3  10–3)(7  10–5) = = 21  10–8 Simplify. = 2.1  101 • 10–8 Write 21 in scientific notation. = 2.1  101 + (– 8) Add exponents of powers with the same base. = 2.1  10–7 Simplify.

Practice Rewrite each expression using each base only once. 5 3 ∙ 5 6 b. (2 4 )( 2 −3 ) c. 7 −3 ∙ 7 2 ∙ 7 6 2. Simplify each expression. 𝑛 2 ∙ 𝑛 3 ∙7𝑛 b. 2𝑦 3 ∙ 7𝑥 2 ∙ 2𝑦 4 c. (𝑚 2 ) 𝑛 −2 7𝑚 3. Simplify the expression. Write the answer in scientific notation. (2.5 x 10 8 )(6 x 10 3 ) 5 9 2 1 =2 7 5 7𝑛 6 28𝑥 2 𝑦 7 7𝑚 3 𝑛 2 1.5 x 10 12