Download 5.4 Dividing Monomials: The Quotient Rule and Integer Exponents

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