Section 1.6Properties of Exponents 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

Negative coefficients: -x4 = -(x4) but (-x)4 = x4

Product x3· x5 = x3+5 = x8

Quotient x6 / x2 = x6-2 = x4

Power (x4)3 = x4·3 = x12

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

Negative Power of Quotients (x3/y5)-1 = y5/x311.6

Product Rule

Can x2x be simplified? x3

Can x5y6 be simplified? no, unlike bases Can a2b7a3 be simplified? a5b7

Can x5+x6 be simplified? no, only products21.6

Examples – Products (-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

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 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 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 A negative exponent means make it the reciprocal 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 (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 Simplifying inside Using exponent ops

2

8

2422254

2

25

24

9

33

9

3

x

y

x

yyx

yx

yx

4

8

161

84

42

4

53

72

16

2

6

3

x

yyx

xy

yx

yx

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Next Time …

1.7 Scientific Notation and 2.1 Graphs

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