11.1Rational Exponents

VocabularyAn exponent is the power that a base is being

raised to…You should know this already.Keep in mind that an exponent counts the

number of copies of the base that are being multiplied together.

When working with exponents, it is often times easier to rewrite a given statement/function using the laws/properties/rules of exponents.

Product RuleWhen multiplying like bases, we can add the

exponents:

Why?

€

am ⋅an = am+n

€

am ⋅an = a ⋅a ⋅...⋅am of these

1 2 4 3 4 ⎛ ⎝ ⎜

⎞ ⎠ ⎟ a ⋅a ⋅...⋅a

n of theses1 2 4 3 4 ⎛ ⎝ ⎜

⎞ ⎠ ⎟

= a ⋅a ⋅a ⋅...⋅am + n all together1 2 4 3 4

= am+n .

Quotient RuleWhen dividing like bases, subtract the powers.

Why?

€

am

an= am−n

€

am

an=a ⋅a ⋅a ⋅a ⋅...⋅a

m of these6 7 4 4 8 4 4

a ⋅a ⋅a ⋅a ⋅...⋅an of these

1 2 4 4 3 4 4

=aa⋅ aa⋅...⋅ a

an of these

1 2 4 3 4 ⋅a ⋅a ⋅...⋅a

m - n of these1 2 4 3 4

m factors all together6 7 4 4 4 8 4 4 4

Power to a Power RuleWhen raising a power to a power, multiply the

powers.

Why?

€

am( )n

= am⋅n

€

am( )n

= am( ) am( )... am( )n of these with each one consisting of m copies of a

1 2 4 4 3 4 4

m copies of a( ) + m copies of a( ) + ... n times results in m ⋅n copies of a.

Product & Quotient Rules (2)

When raising a product or a quotient to a power you can distribute the power.

*NOTE: This cannot EVER be done when there is addition or subtraction involved.

€

ab( )m = ambmab ⎛ ⎝

⎞ ⎠

m

=am

bm

Nth rootThese next two rules are by far the rules that

you will use the most in calculus.Knowing these rules and recognizing that they

can be used, will save you a MAJOR headache.

€

bn = b1n bmn = b

mn

Negative Powers

€

b−n =1bn

Example 1Express using rational exponents:

€

8x 3y 43

€

8x 3y 43 = 8x 3y 4( )1

3

= 81

3 x 3( )1

3 y 4( )1

3

= 2xy4

3 .

Example 2Express the following using radicals:

€

5a( )3

2 2b( )3

4

€

5a( )3

2 2b( )3

4 = 5a( )3 ⋅ 2b( )34

Example 3Simplify:

€

2x( )3 ⋅ 3x 25

4x 5

€

2x( )3 ⋅ 3x 25

4x 5 = 2x( )3 ⋅ 3x 2( )1

5 ⋅4−1x−5

= 23 ⋅31

5 ⋅4−1

unlike bases... can' t combine with the rules

1 2 4 3 4 ⋅ x 3 ⋅x2

5 ⋅x−5

like bases being multiplied,using our product rule

1 2 4 3 4

= 8 ⋅4−1 ⋅31

5 ⋅x 3+25−5

= 2 ⋅31

5 ⋅x−85

Example 4Simplify:

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2xy( )3

2 ⋅ x 4y 3( )3

2

€

2xy( )3

2 ⋅ x 4y 3( )3

2 = 23

2 ⋅x3

2 ⋅y3

2 ⋅ x 4( )3

2 y 3( )3

2

= 23

2 ⋅x3

2 ⋅x12

2 ⋅y3

2 ⋅y9

2

= 23

2 ⋅x15

2 ⋅y 6

Example 5Write without using fractions:

€

2x 3 −

5x 2 + 3x 4

€

2x 3 −

5x 2 + 3x 4 = 2x −3 − 5x −2 + 3x 4

HomeworkPg. 602 # 14-52 [3 each Section]