Power of a PowerPower of a PowerFinding powers of numbers with Finding powers of numbers with exponentsexponents

(x(xmm))nn = x = xmnmn

SimplifySimplify

(2(233))22

This means 2This means 233*2*23 3

2233*2*233 = (2*2*2)*(2*2*2)=2 = (2*2*2)*(2*2*2)=266

SimplifySimplify

(4(422))33

This means 4This means 422*4*42 2 *4*422

4422*4*422*4*422 = (4*4)*(4*4)*(4*4)=4 = (4*4)*(4*4)*(4*4)=466

How does this work?How does this work?

Look againLook again(4(422))3 3 = 4= 466

(2(233))22 =2 =266

How do the exponents 2 How do the exponents 2 and 3 relate to the and 3 relate to the exponent 6?exponent 6?

Let’s look at some moreLet’s look at some more

(3(33)43)4 = (3*3*3)*(3*3*3)*(3*3*3)*(3*3*3) = (3*3*3)*(3*3*3)*(3*3*3)*(3*3*3) (3(33)43)4 =3 =3????

33(3x4)(3x4) = = 331212

As you can see (3As you can see (33)43)4 shows 3 multiplied by shows 3 multiplied by itself 12 times.itself 12 times.

(3(33)4 3)4 = 3= 3(3*4)(3*4)=3=31212

Let’s try some using the Let’s try some using the Power of Powers Power of Powers PropertyProperty

The Power of Powers Property The Power of Powers Property states that when you have a number states that when you have a number to a certain power raised to another to a certain power raised to another power, you multiply the exponents.power, you multiply the exponents.

ExamplesExamples(3(333))4 4 = 3= 31212

(8(822))5 5 = 8= 81010

(9(911))4 4 = 9= 944

Try someTry some

(2(23)4 3)4 = ?= ?

(10(103)2 3)2 = ?= ?

(p(p2)5 2)5 = ?= ?

(x(xm)3 m)3 = ?= ?

Go to the next slide when you have the Go to the next slide when you have the solutions to check your work.solutions to check your work.

Power of PowersPower of Powers

(2(23)4 3)4 = 2= 21212

(10(103)2 3)2 = 10= 1066

(p(p2)5 2)5 = p= p1010

(x(xm)3 m)3 = x= x3m3m

Raise a monomial to a Raise a monomial to a powerpower

(xy)(xy)22 = xy*xy = x*x*y*y = x = xy*xy = x*x*y*y = x22yy22

(xy(xy22))2=2=

If you get stuck with powers of powers, If you get stuck with powers of powers, try writing out the multiplication of try writing out the multiplication of numbers and variables.numbers and variables.

(x*y*y)* (x*y*y)(x*y*y)* (x*y*y) = x*y*y*x*y*y= x*y*y*x*y*y = x*x*y*y*y*y = x= x*x*y*y*y*y = x22yy44

Try someTry some

(xy)(xy)2 2 = ?= ?

(xy(xy22))2 2 = ?= ?

((rr22))4 4 = ?= ?

Go to the next slide when you have the Go to the next slide when you have the solutions to check your work.solutions to check your work.

SolutionsSolutions

(x(x11y)y)2 2 = x= x22yy22

(x(x11yy22))2 2 = x= x22yy44

((11rr22))4 4 = = 44rr88

Can you see the power of powers property at Can you see the power of powers property at work?work?

If not, try changing the variables that have no If not, try changing the variables that have no exponent to an exponent of one.exponent to an exponent of one.

{Once again, 1 comes in handy!}{Once again, 1 comes in handy!}

is pi

Let’s take another lookLet’s take another look

(xy)(xy)2 2 =(x=(x11yy11))2 2 = x= x22yy22

(x(x11yy22))2 2 = x= x22yy44

((11rr22))4 4 = = 44rr88

Try some more. Try some more. Use 1 to your advantage Use 1 to your advantage when you can.when you can.

(x(x22y)y)33= (x= (x22yy11))33= x= x(2*3)(2*3)y(y(1*3)1*3)= x= x66yy33

(x(x22yy22zz22))33==

(abcd)(abcd)nn==

(x(x22yy33))55==

SolutionsSolutions

(x(x22yy22zz22))33=x=x2*32*3yy2*32*3zz2*32*3=x=x66yy66zz66

(abcd)(abcd)nn=a=annbbnnccnnddnn

(x(x22yy33))55=x=x2*52*5yy3*5 = 3*5 = xx1010yy15 15

Powers of -1Powers of -1

Write out (-2)Write out (-2)33== (-2)*(-2)*(-2) (-2)*(-2)*(-2)

When the exponent is an odd number, the When the exponent is an odd number, the answer can be negative.answer can be negative.

2 2 2 4 2

SuggestionSuggestion

Once again, the Once again, the suggestion is to write suggestion is to write out the multiplication out the multiplication statements to help you statements to help you solve tricky solve tricky exponential products.exponential products.

SimplifySimplify

(-t)(-t)55=?=?

(-t)(-t)44=?=?

(-5x)(-5x)33=?=?

solutionssolutions

(-t)(-t)55= (-t)= (-t)* * (-t)(-t)* * (-t)(-t)* * (-t)(-t)* * (-t)(-t)

=-t=-t55

(-t)(-t)44=t=t44

(-5x)(-5x)33=(-5x) (-5x) (-5x) = =(-5x) (-5x) (-5x) =

= -5*-5*-5*x*x*x = -125x= -5*-5*-5*x*x*x = -125x33

Negative and Zero Negative and Zero ExponentsExponentsIntegrated II Integrated II

Chapter 9.2Chapter 9.2

Negative Negative Integers do Integers do NOT mean NOT mean negative negative numbersnumbers

Numbers to the Zero Numbers to the Zero PowerPower

Every number to the Zero Every number to the Zero Power, such as 5Power, such as 500 = 1. = 1.

We can use last lesson’s We can use last lesson’s division of powers as a division of powers as a proof.proof.

Using division to proveUsing division to prove Any number divided by itself equals 1.Any number divided by itself equals 1.

Using the Quotient of Powers Property, Using the Quotient of Powers Property, the exponents would be subtracted.the exponents would be subtracted.

665-5 5-5 = 6= 60 0 = 1= 1

4 4 1 5 56 6 1

Negative Negative ExponentsExponents Negative Exponents do Negative Exponents do notnot

mean negative numbers.mean negative numbers.

44-5 -5 ==

33-2 -2 ==

77-4 -4 ==2

1

34

1

7

5

1

4

Solve.Solve.

2

1

d

4

4

1

1

c

c

4

4

c

c

4 3

6 3

d f

d f

4 4c c

4 6 3 3d f 2 0d f

Simplify.Simplify.

bb66*b*b-2-2 =b =b44 = 1 = 1 bb44 b b44

-3y-3y-2-2

-6p-6p-7-7

8a8a44bb77cc-4-4

3a3a66bb-6-6cc-4-4

Simplify.Simplify.

--3y3y-2 = -2 = -3-3

yy22

-6p-6p-7 -7 = = -6-6

pp77

8a8a44bb77cc-4 -4 = = 88 a a4-64-6bb7--67--6cc-4--4 = -4--4 = 88 bb13 13

3a3a66bb-6-6cc-4 -4 3 3 a3 3 a22

****(c(c00=1 which when multiplied is no longer part of the answer.=1 which when multiplied is no longer part of the answer.

Let’s Divide!Let’s Divide!

Dividing MonomialsDividing Monomials

Focus: Quotient of Powers RuleFocus: Quotient of Powers Rule

Quotients of PowersQuotients of Powers

How do I findHow do I find ??

a*a*a*a*a*aa*a*a*a*a*a == a*a*a*aa*a*a*a

aa22

1 1 = = aa22

6

4

a

a

a*a*a*a*a*a = a*a*a*a

Let’s find a different wayLet’s find a different way6

4

a

aIn the previous slide, In the previous slide,

you saw that the you saw that the result of this fraction result of this fraction was awas a22. .

How do 6 and 4 relate How do 6 and 4 relate to two?to two?

Quotient of Powers Quotient of Powers PropertyProperty

For all non-zero For all non-zero numbers, subtract numbers, subtract the exponent of the the exponent of the denominator denominator fromfrom the the numerator when the numerator when the bases are the same.bases are the same.

445-2 5-2 = 4= 433

5

2

4

4

Let’s prove it.Let’s prove it.

445-2 5-2 = 4= 433 5

2

4 102464

4 16

4 4 4 64

Try Some.Try Some.

10

5

2?

2

10

7

3?

3

8

3

5?

5

3

2

2?

2

SolutionsSolutions

2210-5 10-5 = 2= 255

3310-7 10-7 = 3= 333

558-3 8-3 = 5= 555

223-2 3-2 = 2= 21=1=22

3

2

28 4 2

2

Try Some with variables.Try Some with variables.

xxj-1 j-1

xxa+b-ca+b-c

xxm+1-1 m+1-1 = x= xmm

?jx

x

?a b

c

x

x

1

?mx

x

Fun Fun FunFun Fun Fun

2 5

3

4?

2

x y

xy

4 2

5

48?

1.2

a b

ac

Fun Fun FunFun Fun Fun

-2x-2x2-12-1yy5-35-3=-2xy=-2xy22

-40a-40a4-14-1b2cb2c-5-5

==-40a-40a33bb22

cc55

2 5

3

4?

2

x y

xy

4 2

5

48?

1.2

a b

ac

Remember negative Remember negative exponents?exponents?

Any time you have a negative Any time you have a negative exponent, it must be placed exponent, it must be placed in the denominator.in the denominator.

C C -3-3 = =

3

1

c

Try SomeTry Some

45 32

1133

answersanswers4

4

1 15

5 625

33

1 12

2 8

1 1133

133

Combine Combine two two conceptsconcepts33 25a b

a

32

3

Combine two concepts Combine two concepts (answers)(answers)

33 22 2 3 6 65

(5 ) 125a b

a b a ba

3 3

3

2 2 8

3 3 27

Thanks for Thanks for enjoying math!enjoying math!

Nothing will be due Nothing will be due for today’s work.for today’s work.