### 1.6

```Section 1.6
Properties of Exponents
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Why do you need to become Exponent Experts?
Terms & Definitions Base, Exponent, Power
 x to the 5th power
x5 = x · x · x · x · x
Rules for Exponents
4
4
 Negative coefficients: -x = -(x4) but (-x) = x4
 Product
x3· x5 = x3+5 = x8
 Quotient
x6 / x2 = x6-2 = x4
 Power
(x4)3 = x4·3 = x12
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Power of Products (x6 y9)2 = x6·2y9·2 = x12 y18
Power of Quotients (x3/y5)4 = x3·4/y5·4 = x12/y20
Zero
x0 = 1
430 = 1
Negative x-7 = 1 / x7 -1 means Reciprocal
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Negative Power of Quotients
(x3/y5)-1 = y5/x3
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Product Rule
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Can
Can
Can
Can
x2x be simplified? x3
x5y6 be simplified? no, unlike bases
a2b7a3 be simplified? a5b7
x5+x6 be simplified? no, only products
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Examples – Products
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(-2)4 = (-2)(-2)(-2)(-2) = 16
-24 = -(2)(2)(2)(2) = -16
x3x2x7x = x3+2+7+1 = x13
y2y5 = y7
xxx3 = x5
b2cb3 = b5c
x3+x = x3+x
(-5)3 = (-5)(-5)(-5) = -125
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The Quotient Rule
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Example
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What if there are more on the bottom?
x2/x5
1/x3
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When an Exponent is Zero
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Examples – Quotient Rule
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Product is addition – Quotient is subtraction
x5x2 = x5+2 = x7
x5/x2 = x5-2 = x3
You try:
y5/y4 = y x11/x3 = x8 x9/x9 = x9-9 = x0 = 1
x4/y2 = x4/y2 xy3/y = xy2
x2/x8 = x2-8 = x-6 = 1/x6
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Negative Exponents
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Examples – Zero and Negative
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x3 = xxx x2 = xx x1 = x x0 = 1
Think: Only the coefficient remains
60 = 1 2y0 = 2 (3y2z)0 = 1 (x+3)0 = 1 -y0 = -1
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A negative exponent means make it the reciprocal
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6-1 = 1/6 2y-1 = 2/y (3y2)-1 = 1/(3y2) -y-1 = -1/y
2-3 = 1/23 = 1/8 (x+3)-2 = 1/(x+3)2
(3/7)-1 = 7/3 (x/3)-2 = (3/x)2 = 9/x2
x-3/ x-7 = x-3-(-7) = x-3+7 = x4
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The Power Rule
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The Power Rule for Products & Quotients
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Examples –Powers
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(y2)5 = y10
(x2y)3 = x6y3
(bb2b3)4 = b24
(2x4)3 = (2x4)(2x4)(2x4) = 23x4·3 = 8x12
(-2x4)3 = (-2x4)(-2x4)(-2x4) = (-2)3x4·3 = -8x12
(⅓a3b)2 = (⅓a3b)(⅓a3b) = (⅓)2a3·2b1·2 = (a6b2)/9
-(⅓a3b)2 = -(⅓a3b)(⅓a3b) = -(⅓)2a3·2b1·2 = -(a6b2)/9
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Serious Examples
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Simplifying inside
  3x y 


5 2 
  9x y 
2
x


2
4
4 5
y
3
2
2  2
Using exponent ops
 3x y 
 3 5 
 6x y 
2
 y 
   
  3x 
4
7
4
4
 xy 

 
 2 
x 4 y 8
16 y 8
 1
 4
x
16
2
2
y8
 2
9x
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Next Time …
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1.7 Scientific Notation and
2.1 Graphs
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```