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5.4 Dividing Polynomials: The Quotient Rule and Integer Exponents Learning Objectives: 1. Exponential Properties. 2. Simplify using Exponential Properties. 3. Simplify exponential expressions using the Laws of Exponents. 1. Exponential Properties. Exponential Properties: If x is any nonzero real number, and m and n are natural numbers, then 1. Product Rule: 2. Quotient Rule: xm ⋅ xn = xm = xn ⎛x⎞ ⎜⎜ ⎟⎟ = ⎝ y⎠ 2. − x0 = 2. 1 = x −n m 3. Quotient to a Power Rule: 4. Zero as an Exponent: 1. x0 = 3. (− x )0 = 3. ⎛x⎞ ⎜⎜ ⎟⎟ ⎝ y⎠ 5. Negative Exponents: 1. x −n = −m = 2. Simplify using Exponential Properties Example 1. Use the Quotient Rule to simplify. 310 = 37 1. x14 = x8 2. 45 y 12 = 15 y 4 3. − 6a 3 b12 = 8a 4 b 6 4. 5. − 8 x 3 y 5 ÷ −2 xy 2 = 1 Example 2. Use the Zero Product Rule to simplify. 1. 40 = 2. − 150 = 3. − 18 x 0 = Example 3. Use the Negative Exponent Rule to simplify. Write answers with positive exponents only. 1. 3−3 = 2. 8 y −2 = 3. ⎛4⎞ ⎜ ⎟ ⎝ 3⎠ −2 = 4. − 16 = x −5 Example 4. Simplify. Write answers with positive exponents only. 1. a4 = a −3 2. x −10 = x −5 3. 4. 3x 2 y 3 z 4 = − 9 x 5 y 2 z −2 (15x 2 y −1 )(2 x −3 y −4 ) = 2 5. ⎛ 2 x −2 ⎞ ⎜⎜ ⎟ −3 ⎟ 5 xy ⎝ ⎠ −2 = −4 6. 2 −2 ⎛ a ⎞ ⎛ 3a ⎞ ⎜ ⎟ ⋅ ⎜⎜ 4 −4 ⎟⎟ = ⎝ 2b ⎠ ⎝ 6a b ⎠ −4 7. ⎛ −2a 2b −2 c −2 ⎞ ⎜ ⎟ = −3 −4 3 b c ⎝ ⎠ 8. 2−2 2−3 + = 5−1 3−1 3 9. ( 3x y ) 4 3 2 −3 ⎛ −3x 4 y ⎞ ⎜ −2 −1 ⎟ = ⎝ 2x y ⎠ −2 −1 ⎛ 5a −2 ⎞ ⎛ 5 ⎞ 10. ⎜⎜ 3 ⎟⎟ + ⎜ 4 6 ⎟ ⎝a b ⎠ ⎝ b ⎠ 0 ⎛ −1.2 x 4 y10 ⎞ 11. ⎜ = −5 −3 ⎟ ⎝ 6x y ⎠ 4