Exponent Laws II

Topic 2.5

Power As a Repeated Multiplication

As a Productof Factors

As a Power As a Product of Powers

(24)3 24 x 24 x 24 (2)(2)(2)(2) x(2)(2)(2)(2) x(2)(2)(2)(2) x

212

(32)4

[(-43)2]

(2x5)3 (2x5)x(2x5)x(2x5) 2x5x2x5x2x5 23 x 53

(3x4)2

(4x2)5

OVERVIEWOVERVIEW

POWER OF A POWERPOWER OF A POWER

(32)4

= 32 X 32 X 32 X 32 What do you do with the exponents of like bases when

they are multiplied together? (Last section)What do you do with the exponents of like bases when

they are multiplied together? (Last section)

ADD!!! = 32+2+2+2

= 38

This answer is the same as multiplying the exponents together.

This answer is the same as multiplying the exponents together.

=32x4

=32x4

POWER OF A POWERPOWER OF A POWER

Proper Definition

(na)b = naxb

for any n, a, and b in the real numbers.

Why don’t we just do this?Why don’t we just do this?

(32)4 = (9)4 = 9 x 9 x 9 x 9 = 6561Because sometimes we could get really difficult numbers.Because sometimes we could get really difficult numbers.

Why don’t we just do this?Why don’t we just do this?Because sometimes we could get really difficult numbers.Because sometimes we could get really difficult numbers.

(912)4 = (282429536481)4

(282429536481)x(282429536481)x(282429536481)x(282429536481)

This is way harder than just doing this:This is way harder than just doing this:

(912)4 = 912 x 4 = 948

Exponent Law for POWER OF A POWER

To find a power of a power, MULTIPLY the exponents!

To find a power of a power, MULTIPLY the exponents!

(62)7

= 62x7

=

[(-7)3]2

= (-7)3x2

=

-(24)5

= -(24x5)

=

Write each as a power.

POWER OF A PRODUCTPOWER OF A PRODUCT

=(2x3)3

=(2x3)(2x3)(2x3)

Remember, you can multiply in any order, so group the same numbers

Remember, you can multiply in any order, so group the same numbers

=2x2x2x3x3x3

=23 x 33

Simplify, then evaluate.

=216

Is there another way to figure this

out?

Is there another way to figure this

out?

To find a power of product, DISTRIBUTE the exponents to

each base!

To find a power of product, DISTRIBUTE the exponents to

each base!

POWER OF A PRODUCTPOWER OF A PRODUCTThese two methods will give you the same answer.

(2x3)3

=23 x 33

=216

Method 1 Method 2

(2x3)3

=(6)3

=216

Again the numbers can get messy on you, and when you start using variables only method 1 will work

POWER OF A PRODUCTPOWER OF A PRODUCT

Proper Definition

(m x n)a = ma x na

for any m, n, and a in the real numbers.

POWER OF A QUOTIENTPOWER OF A QUOTIENT3

5

6 5 5 5

6 6 6

3

3

5

6

442

18

To find a power of QUOTIENT, DISTRIBUTE the exponents to each

base, then evaluate (if you are asked to!).

To find a power of QUOTIENT, DISTRIBUTE the exponents to each

base, then evaluate (if you are asked to!).

125

216

47

3

Simplify First!Simplify First!

4

4

7

3

5 5 5

6 6 6

POWER OF A QUOTIENTPOWER OF A QUOTIENT

Proper Definition

for any m, n, and a in the real numbers.

a a

a

m m

n n

Power of a power

Power of a product

Power of a power

(43)5 = 43x5 = 415

(3x8)4 = 34 x 84

3 3

3

6 6

7 7

ASSIGNMENTASSIGNMENT

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