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Exponents

{ 35Power

base

exponent

3 3 means that is the exponentialform of tExample:

he number125 5 5

.125=

53 means 3 factors of 5 or 5 x 5 x 5

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The Laws of Exponents:#1: Exponential form: The exponent of a power indicates how many times the base multiplies itself.

3Example: 5 5 5 5= ⋅ ⋅

n factors of x

#2: Multiplying Powers: If you are multiplying Powers with the same base, KEEP the BASE & ADD the EXPONENTS!

m n m nx x x +⋅ =

So, I get it! When you multiply Powers, you add the exponents! 512

2222 93636

=

==× +

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#3: Dividing Powers: When dividing Powers with the same base, KEEP the BASE & SUBTRACT the EXPONENTS!

mm n m n

n

x x x xx

−= ÷ =

So, I get it!

When you divide Powers, you subtract the exponents!

16

2222 4262

6

=

== −

Try these:

=× 22 33.1=× 42 55.2

=× 25.3 aa

=× 72 42.4 ss

=−×− 32 )3()3(.5

=× 3742.6 tsts

=4

12

.7ss

=5

9

33.8

=44

812

.9tsts

=54

85

436.10

baba

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=× 22 33.1=× 42 55.2=× 25.3 aa

=× 72 42.4 ss

=−×− 32 )3()3(.5

=× 3742.6 tsts

8133 422 ==+

725 aa =+

972 842 ss =×× +

SOLUTIONS

642 55 =+

243)3()3( 532 −=−=− +

793472 tsts =++

=4

12

.7ss

=5

9

33.8

=44

812

.9tsts

=54

85

436.10

baba

SOLUTIONS

8412 ss =−

8133 459 ==−

4848412 tsts =−−

35845 9436 abba =×÷ −−

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#4: Power of a Power: If you are raising a Power to an exponent, you multiply the exponents!

( )nm mnx x=So, when I take a Power to a power, I multiply the exponents

52323 55)5( == ×

#5: Product Law of Exponents: If the product of the bases is powered by the same exponent, then the result is a multiplication of individual factors of the product, each powered by the given exponent.

( )n n nxy x y= ⋅So, when I take a Power of a Product, I apply the exponent to all factors of the product.

222)( baab =

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#6: Quotient Law of Exponents: If the quotient of the bases is powered by the same exponent, then the result is both numerator and denominator , each powered by the given exponent.

n n

n

x xy y

⎛ ⎞=⎜ ⎟

⎝ ⎠So, when I take a Power of a Quotient, I apply the exponent to all parts of the quotient. 81

1632

32

4

44

==⎟⎠

⎞⎜⎝

⎛

Try these:

( ) =523.1( ) =43.2 a

( ) =322.3 a

( ) =23522.4 ba

=− 22 )3(.5 a

( ) =342.6 ts

=⎟⎠

⎞⎜⎝

⎛5

.7ts

=⎟⎟⎠

⎞⎜⎜⎝

⎛2

5

9

33.8

=⎟⎟⎠

⎞⎜⎜⎝

⎛2

4

8

.9rtst

=⎟⎟⎠

⎞⎜⎜⎝

⎛2

54

85

436.10

baba

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( ) =523.1( ) =43.2 a

( ) =322.3 a

( ) =23522.4 ba

=− 22 )3(.5 a

( ) =342.6 ts

SOLUTIONS

10312a

6323 82 aa =×

6106104232522 1622 bababa ==×××

( ) 4222 93 aa =×− ×

1263432 tsts =××

=⎟⎠

⎞⎜⎝

⎛5

.7ts

=⎟⎟⎠

⎞⎜⎜⎝

⎛2

5

9

33.8

=⎟⎟⎠

⎞⎜⎜⎝

⎛2

4

8

.9rtst

=⎟⎟⎠

⎞⎜⎜⎝

⎛2

54

85

43610

baba

SOLUTIONS

( ) 62232223 8199 babaab == ×

2

8224

rts

rst

=⎟⎟⎠

⎞⎜⎜⎝

⎛

( ) 824 33 =

5

5

ts

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#7: Negative Law of Exponents: If the base is powered by the negative exponent, then the base becomes reciprocal with thepositive exponent.

1mmxx

− =So, when I have a Negative Exponent, I switch the base to its reciprocal with a Positive Exponent.

Ha Ha!

If the base with the negative exponent is in the denominator, it moves to the numerator to lose its negative sign!

9331

1251

515

22

33

==

==

−

−

and

#8: Zero Law of Exponents: Any base powered by zero exponent equals one.

0 1x =

1)5(

1

15

0

0

0

=

=

=

aandaand

So zero factors of a base equals 1. That makes sense! Every power has a coefficient of 1.

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Try these:

( ) =022.1 ba=× −42.2 yy

( ) =−15.3 a

=×− 72 4.4 ss

( ) =−− 4323.5 yx

( ) =042.6 ts

=⎟⎟⎠

⎞⎜⎜⎝

⎛−122.7

x

=⎟⎟⎠

⎞⎜⎜⎝

⎛−2

5

9

33.8

=⎟⎟⎠

⎞⎜⎜⎝

⎛−2

44

22

.9tsts

=⎟⎟⎠

⎞⎜⎜⎝

⎛−2

54

5

436.10baa

SOLUTIONS

( ) =022.1 ba

( ) =−15.3 a

=×− 72 4.4 ss

( ) =−− 4323.5 yx

( ) =042.6 ts

1

5

1a

54s

( ) 12

81284

813

yxyx =−−

1

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=⎟⎟⎠

⎞⎜⎜⎝

⎛−122.7

x

=⎟⎟⎠

⎞⎜⎜⎝

⎛−2

5

9

33.8

=⎟⎟⎠

⎞⎜⎜⎝

⎛−2

44

22

.9tsts

=⎟⎟⎠

⎞⎜⎜⎝

⎛−2

54

5

436.10baa

SOLUTIONS

44 1 xx

=⎟⎠

⎞⎜⎝

⎛−

( ) 8824

3133 == −−

( ) 44222 tsts =−−−

2

101022

819

abba =−−

6.4 Rational Exponents

Fraction Exponents

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Radical expression and Exponents

By definition of Radical Expression.

The index of the Radical is 3.

51251255 33 == so

How would we simplify this expression?

What does the fraction exponent do to the number?

The number can be written as a Radical expression, with an index of the denominator.

21

9

=2 9

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The Rule for Rational Exponents

46464 331

1

==

= nn bb

Write in Radical form

=

=

21

61

m

a

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Write each Radical using Rational Exponents

=

=

w

b5

What about Negative exponents

Negative exponents make inverses.

71

49

14921

21

==−

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What if the numerator is not 1

For any nonzero real number b, and integer m and n

Make sure the Radical express is real, no b<0 when n is even.

( )mnn mnm

borbb =

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Simplify each expression

No fraction in the denominators.

32

74

71

−

⋅

x

yy

Simplify

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6.4 Rational exponents

Solving for the base in a power

• Solve for x. assume x is positive

a. 𝑥" = 16 b. 𝑥&' = 27

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Solving a financial problem

• Under annual compounding, a principal of $700 grows to$900 in 5 years. Determine the annual interest rate. Using the following formula

𝐴 = 𝑝(1 + 𝑖)0

• HW 6.3 page 362 # 1,2,3,6 every other one, 7, 10(a,b), 13

• HW 6.4 page 369 # 1, 2*, 3, 4, 5*, 6(a,b)

• HW 6.5 page 376 # 2, 3, 8, 11

* Every other one