Warm upUse the laws of exponents to simplify the following.

Answer should be left in exponential form.

1 2 22 3.( )( )

2 43 2.( )

35 2.

4 6 72 3 5.( )

5130.

66

6

3

7.

The laws you used in the warm up with integer bases apply to all bases, even variable bases.

The laws also apply for all types of exponents, including fractions and decimals.

Let’s review the rules and see them applied.

Labeling an exponential expression

• The expression is written and read as X to the 5th power.

• X is called the base of the expression.• 5 called a power or exponent for the

expression.• The exponent, 5, tells us that we want X

multiplied with its self 5 times• Power or exponential form is X5 • x x x x x is Expanded form

X5

PracticeIdentify the base of

Identify the exponent of

Is in exponential or expanded form?

ZERO POWER RULEAny base(s) raised to the zero power will

always equal 1.Examples 1234 1

23 1

2

71

0

2 8 9 0

16

4

0

FHG

IKJ

( ).w gy

x

x

Practice

Product of Powers

When two bases are multiplied we add the exponents of the bases.

Examples

If there are numbers in the expression we can multiple them.

Example

( )( )w w w w2 3 2 3 5

( )( )( ) ( )( )my y m m y m y m y13

23 4 1 4 1

323 5 1 5

( )( ) ( )( )3 5 3 5 154 6 4 6 10r r r r

Practice

Power of a PowerWhen we have an exponent raised to an exponent

we multiple the exponents.Example

If there are numbers or more than one variable, inside the parenthesis, they all get raised to the outside power. When we have we first simplify by ‘distributing’ the outside exponent inside, and then, since the two groups are multiplied, we added the exponents of like bases.

Examples

( )a a3 3 9

( )2 2 2 84 3 1 3 4 3 3 12 12a a a a

( )a g a g a g8 3 2 8 2 3 2 16 6

( ) ( ) ( ) ( ) ( )

1 1 14 2 2 8 84 2

j j j j

( ) ( ) ( )(( ) ))2 2 1 85 3 3 2 3 15 3 2 6 21 3y m y y m y y m

Warm up 3-2Use the laws of exponents to simplify the following.

Answer should be left in exponential form.

1 1 2 3.( )( )/x x

2 3 4.( )y

3 6.z

4 2 6 3 5.( )h j

54

5 13 5 3 0.( )/

x y

68 2 1

7 9 4.x y z

x y z

Quotient of PowersWhen dividing with exponents we subtraction

the exponents of common bases. Examples

If there is no other base for you to divide with it is kept in the same place.

Examples

v w

v wv w

5 8

3 32 5r

rr

5

23

33

7

34y

yy

( )e w

w e

e w

w ee w w

2 3

2 6

6 3

2 60

44

2 2

2 62 4 1e w

w ee w

g w

wg w

2 8

32 5

Practice

Power of a QuotientWhen we raising a fraction to a power, we can

rewrite the fraction by raising everything on top by the outside exponent and everything on bottom to the outside exponent.

Examples

w

w

w

w

w

ww

2 2 2 2

2

4

22F

HGIKJ

( )

( )

2 2 2 88

4

3 8 3

4 3

3 24

3 12

12

3

w

gw

w

gw

w

g w

w

g

FHGIKJ

( )

( )

Practice

Negative ExponentsIf we have a negative exponent we can write it as a

positive by taking the reciprocal.If the negative exponent is on the top of a fraction

we can write it positive by simply moving it to the bottom of the fraction.

If the negative exponent is on the bottom of a fraction we can write it positive by moving it to the top of the fraction.

Examples

xx

22

1

16

6

gg 3

36

6

gg

3

2

3

27

7

g

g

3

2

3

2

2

5 5 2

e

g g e

Practice

Fraction powers can be written as radicals, roots.

Practice