Power of a PowerPower of a Power

Finding powers of numbers Finding powers of numbers with exponentswith exponents

(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 How do the exponents 2 and 3 relate to the 2 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????

►3x4 = 123x4 = 12►As you can see (3As you can see (33)43)4 shows 3 multiplied shows 3 multiplied

by itself 12 times.by itself 12 times.►(3(33)4 3)4 = 3= 33*43*4=3=31212

Let’s try some using the Power of Let’s try some using the Power of Powers PropertyPowers Property

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

►ExamplesExamples(3(33)4 3)4 = 3= 31212

(8(82)5 2)5 = 8= 81010

(9(91)4 1)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

Lesson 8.2 Part TwoLesson 8.2 Part Two

Raise a monomial to a powerRaise a monomial to a power

(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 If you get stuck with powers of powers, try writing out the powers, try writing out the multiplication of numbers and multiplication of numbers and variables.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(xy2)2 2)2 = ?= ?

(πr(πr2)4 2)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(x11yy)2 )2 = x= x22yy22

(x(x11yy2)2 2)2 = x= x22yy44

(π(π11rr2)4 2)4 = π= π44rr88

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!}

Let’s take another lookLet’s take another look

(x(x11yy)2 )2 = x= x22yy22

(x(x11yy2)2 2)2 = x= x22yy44

(π(π11rr2)4 2)4 = π= π44rr88

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= (x22y1)y1)33= x= x2*32*3*y*y11**33= 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, When the exponent is an odd number, the answer can be negative.the answer can be negative.

[(-2)*(-2)]*(-2)=[(-2)*(-2)]*(-2)=

[+4] * (-2) = -8[+4] * (-2) = -8

SuggestionSuggestion

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