Math-1010

Lesson 1-9

Properties of Exponents

Properties of Exponents

What is a power?

Power: An expression formed by repeated

multiplication of the base.

43xCoefficient Base

Exponent

The exponent applies to the number or variable

immediately to its left, not to the coefficient !!!

No Exponent? x3 113 xUsually, we don’t write the exponent ‘1’ (saves ink).

No Coefficient?3x 3*1 x 31 *1 x

Usually, we don’t write the coefficient ‘1’ (saves ink).

Negative?2x 2*)1( x 21 *)1( x

Usually, we don’t write the coefficient ‘-1’, we just

put the “negative symbol” (saves ink).

4x

Base Exponent

Factor: a number that is being multiplied.

means “base x used as a factor 4 times”

xxxxx ***4

Product: the result (“answer”) when multiplying.

Power: is repeated multiplication

xxxxx ***4

multiplication: is repeated addition

xxxx 3

How can you tell which definition to use?

Use the “context” of the problem

xx 43 (adding two terms)

)()( xxxxxxx

xxx 743

How can you tell which definition to use?

Use the “context” of the problem

22 32 xx (adding two terms)

)()( 22222 xxxxx

222 532 xxx

How can you tell which definition to use?

Use the “context” of the problem

32 * xx (multiplying two terms)

)**)(*( xxxxx

532 * xxx

Exponents

?3 2 x

?)3( 2 x

?)3(4 2 x

23x

)3)(3( xx

There’s no way to

simplify this anymore.

29x

)9(4 2x

In PEMDAS, exponents occur before

multiplication.

236x

2*9*4 x

Exponents

?2

2

x

?3

23

x

22

xx Remember how to

multiply fractions?4

2x

xxx 3

2

3

2

3

2327

8

x

Simplify2)4( y

2)5(2 x3

2

x

42

x216y

250x

4

16

x

8

3x

Product of Powers Property

))(( 32 xx

)**)(*())(( 32 xxxxxxx

5x

This is ‘x’ used as a factor how many times?

‘x’ used as a factor five times

3232 xxxWhen you multiply powers having the

same base, you add the exponents.

Power of a Power Property

)*)(*)(*()( 32 xxxxxxx

32 )(x6x

63*232)( xxx

32 )(x

This is ‘x’ used as a factor how many times?

‘x’ used as a factor six times

When you raise a power to another power

(“power of a power”)

you multiply the exponents.

Power of a Product Property 2)(xy

))(( xyxy yxyx *** yyxx ***

22 yx

This makes it seem like you can “distribute” in

the exponent. This only works with the power of a product!!

mmm yxxy )(

222)( yxyx

You must “FOIL” the power of a suml!!

))(()( 2 yxyxyx

22 2 yxyx

Combination of 1. Power of a Product2. Power of a Power

243 )3( yx

2431 )3( yx

8623 yx

‘x’ is a number, we just don’t know what it is. You treat all

numbers the same (whether they are variables or constants).

cmbmammcba yxyx 3)3(

Constants (integer, etc.) usually have an exponent of ‘1’.

512x

))((*4*3 32 xx

?)4(3 32 xx

You can re-arrange the order of

multiplication.

Coefficients of the powers are handled

separately from the base and the exponent.

Simplify:

)2)(5( 32 xx

3

2

1)2( xx

510x

4x

52 )2( x

43 2*)(5 xx

1032x

710x

35 )2( y158y

What is the difference between?

44 x)( andx

2332 )( and )( xx

4334 and xxxx

)1)(1( and )1( 2 xxx

Watch the negatives! 243 )( yx2431 ))1(( yx

862)1( yx This way you will be able to tell if the

simplified version is positive or negative. 86 yx

Turn negative signs into multiplication by -1.

362 )2( yx

3621)2( yx

18632 yx

Negative coefficients have an exponent of ‘1’.

1868 yx

A negative number raised to an odd

exponent remains negative.

Your Turn: simplify

534 )(2 xm

342 )2( zyx 31268 zyx

15202 xw

432 )2(3 yzx 1248x

Negative Exponent Property

2x

“Grab and drag”

1

*1 2

x

2

1

x

When you “Grab and drag” the base and its

exponent across the “boundary line” between

numerator and denominator, you just change the

sign of the exponent.

Why do we call this the negative exponent property?

22 yx2

2

y

xExamples:

2

3

1

x 6

1

x

6x6

1

x

Negative Exponent Property

24 x

Possible errors

1

*4 2

x

2

4

x

When you “Grab and drag” the base and its exponent across

the “boundary line” between numerator and denominator,

you just change the sign of the exponent.

DO NOT GRAB the coefficient!2

2

4

1

1

*4

x

x

Zero Exponent Property

120

1)2( 0 x

21*22 0 x

310 1000

210 100

110 10

010 1

Any base raised to the zero power simplifies to one.

Combination: (1) Negative Exponent, (2) Product of Powers, (3) Power of a Power, (4) Power of a Quotient

2

4

2

2

3

yx

x2

11

61

2

3

y

x2

42

2

3

y

xx2

6

2

3

y

x

2*12*1

2*62*1

2

3

y

x

22

122

2

3

y

x

2

12

4

9

y

x

Your Turn:52 )( w

2

32

2

3

2

zy

x

2332

1 x

1

2

4

3

5

y

x

10

1

w

62

9

x

4

64

4

9

x

zy

245

3

yx

Examples:

172

1032

yx

x

17

21032

y

xx

“Grab and drag”

17

21032

y

x

Product of powers: add

the exponents of same

based powers

17

832

y

x

Your Turn:

5

3

4

2

x

x

x

x

2

)2(9 4

3

42

2

)(

x

x

3

432

2

)2(

yx

xy

22

1

x

372x

52

1

x

9

78

x

y

yx

x4

2

2

3

Do you “grab and drag (up or down)??

yx

x4

2

2

3

y

xx

2

3 42

y

x

2

3 42

Product of powers property: add the

exponents of like-based powers

y

x

2

3 6

yxx 242

3

Product of powers property: add the

exponents of like-based powers

yx 242

3

yx 62

3

Make sure when you’re all done, there are

NO NEGATIVE EXPONENTS remaining.

y

x

2

3 6

It doesn’t matter!!!!