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Lesson 1-9

Powers and Laws of Exponents

Location of Exponent

An exponent is a little number high An exponent is a little number high and to the right of a regular or base and to the right of a regular or base number.number.

3 4Base

Exponent

Definition of Exponent

An exponent tells how many times An exponent tells how many times a number is multiplied by itself. a number is multiplied by itself.

3 4Base

Exponent

What an Exponent Represents

An exponent tells how many times An exponent tells how many times a number is multiplied by itself.a number is multiplied by itself.

34= 3 x 3 x 3 x 3

How to read an Exponent

This exponent is read This exponent is read three to three to the fourth power.the fourth power.

3 4Base

Exponent

How to read an Exponent

This exponent is read This exponent is read three to three to the 2the 2ndnd power power oror three three squared.squared.

3 2Base

Exponent

How to read an Exponent

This exponent is read This exponent is read three to three to the 3rd power the 3rd power oror three three cubed.cubed.

3 3Base

Exponent

Read These Exponents

3 2 6 72 3 5 4

What is the Exponent?

2 x 2 x 2 = 23

What is the Exponent?

3 x 3 = 3 2

What is the Exponent?

5 x 5 x 5 x 5 = 54

What is the Base and the Exponent?

8 x 8 x 8 x 8 = 8 4

What is the Base and the Exponent?

7 x 7 x 7 x 7 x 7 =7 5

What is the Base and the Exponent?

9 x 9 = 9 2

How to Multiply Out an Exponent to Find the

Standard Form

= 3 x 3 x 3 x 33

927

81

4

What is the Base and Exponentin Standard Form?

4 2= 16

What is the Base and Exponentin Standard Form?

2 3= 8

What is the Base and Exponentin Standard Form?

3 2= 9

What is the Base and Exponentin Standard Form?

5 3= 125

Exponents Are Often Used inArea Problems to Show the

Feet Are Squared

Length x width = areaA pool is a rectangleLength = 30 ft.Width = 15 ft.Area = 30 x 15 = 450 ft. 2

15ft.

30ft

Exponents Are Often Used inVolume Problems to Show the

Centimeters Are CubedLength x width x height = volumeA box is a rectangleLength = 10 cm.Width = 10 cm.Height = 20 cm.Volume =

20 x 10 x 10 = 2,000 cm.3

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10

10

Here Are Some AreasChange Them to Exponents

40 feet squared = 40 ft.56 sq. inches = 56 in.38 m. squared = 38 m.56 sq. cm. = 56 cm.

2

2

2

2

Here Are Some VolumesChange Them to Exponents

30 feet cubed = 30 ft.26 cu. inches = 26 in.44 m. cubed = 44 m.56 cu. cm. = 56 cm.

3

3

3

3

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Law of Exponents for Law of Exponents for MultiplicationMultiplication

To multiply two powers that have the same base, To multiply two powers that have the same base, keep the base and add the exponents.keep the base and add the exponents.

xa • xb = xa+b

Examples : 42 • 43 = 45

95 • 98 = 913

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Law of Exponents for DivisionLaw of Exponents for Division

To divide two powers that have the same base, keep To divide two powers that have the same base, keep the base and subtract the exponents.the base and subtract the exponents.

xa ÷ xb = xa-b

Examples : 75 ÷ 73 = 72

28 ÷ 22 = 26

Remember that division can also be written vertically:

Now here’s a harder one!

25

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Law of Exponents for ZeroLaw of Exponents for Zero

A number to the zero power always equals 1.A number to the zero power always equals 1.

Simplify: 36

36

3•3•3•3•3•33•3•3•3•3•3

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But what happens if you add But what happens if you add or subtract the exponents and or subtract the exponents and you get a negative number ?you get a negative number ?

First of all, there is no crying in First of all, there is no crying in math! Second, we have a law math! Second, we have a law

for that too! It’s called the for that too! It’s called the Negative Rule! Let me tell you Negative Rule! Let me tell you

all about it…all about it…

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Negative RuleNegative Rule

Any non-zero number raised to a negative power Any non-zero number raised to a negative power equals its reciprocal raised to the opposite equals its reciprocal raised to the opposite positive power. positive power.

WHAT!WHAT!!!

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……Negative Rule Negative Rule Remember that a reciprocal is the multiplicative Remember that a reciprocal is the multiplicative

inverse. In simple terms, flip the fraction! The inverse. In simple terms, flip the fraction! The reciprocal of is .reciprocal of is .

If we apply the negative rule (If we apply the negative rule (Any non-zero number raised Any non-zero number raised to a negative power equals its reciprocal raised to the opposite positive to a negative power equals its reciprocal raised to the opposite positive

powerpower) then, ) then, In this example, the negative in front of the four remains. Only the negative of the exponent is effected.

A non-zero raised to a negative power =

The reciprocal raised to the opposite power

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Power RulePower RuleWhen raising a power to a power, keep the base When raising a power to a power, keep the base

and multiply the exponents.and multiply the exponents.

(xa)b = xa•b

Examples: (24)3 = 212

(x3)5 = x15

Let me jot this down.

Oh yes, I got it now!

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Product to a Power RuleProduct to a Power RuleA product raised to a power is equal to each base A product raised to a power is equal to each base

in the product raised to that exponent.in the product raised to that exponent.

(7• 3)2 = 72 •32 = 49 • 9 =441 (x3y2)5 = x15y10

(2x2yz-3)-4 = 2-4x-8 y-4 z12 = =

(x • y)2 = x2y2

Here’s one where the variables have exponents

Here’s one where the product is raised to a negative power!

Examples:

Tricky, trickier, trickiest – But I think I got it!

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Quotient to a Power RuleQuotient to a Power RuleA quotient raised to a power is equal to each A quotient raised to a power is equal to each

base in the numerator and denominator raised base in the numerator and denominator raised to that exponent.to that exponent.

Examples:

…and this is the last law! 3

2

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Why does anything to theWhy does anything to the zero power equal 1? zero power equal 1?

22 = 2 x 2 = 4

21 = 2 = 2

23 = 2 x 2 x 2 = 8

24 = 2 x 2 x 2 x 2 = 16

25 = 2 x 2 x 2 x 2 x 2 = 32

Division is a good way of showing how this works:

Take the product for 25 and divide it by 2. 32 ÷ 2 = 16 and 16 = 24

Now take that answer, 16, which is the standard form of 24, and divide it by 2.

16 ÷ 2 = 8 and 8= 23

Now take that answer, 8, which is the standard form of 23, and divide it by 2.

8 ÷ 2 = 4 and 4= 22

Now take that answer, 4, which is the standard form of 22, and divide it by 2.

4 ÷ 2 = 2 and 2 = 21

Now take that answer, 2, which is the standard form of 21, and divide it by 2.

2 ÷ 2 = 1 AND 1 = 20

20 = 1 = Really!

THIS WORKS FOR ALL NUMBERS – CLICK THIS WORKS FOR ALL NUMBERS – CLICK HERE TO SEE ONE MORE EXAMPLE! TO SEE ONE MORE EXAMPLE!

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52 = 5 X 5= 25

51 = 5 = 5

50 = 1

53 = 5 X 5 X 5= 125

54 = 5 X 5 X 5 X 5 = 625

55 = 5 X 5 X 5 X 5 X 5 = 3125 Take the product for 55 and divide it by 5

3125 ÷ 5 = 625 and 625 = 54

Now take that answer, 625, which is the standard form of 54, and divide it by 5

625 ÷ 5 = 125and 125 = 53

Now take that answer, 125, which is the standard form of 53, and divide it by 5

125 ÷ 5 = 25and 25 = 52

Now take that answer, 25, which is the standard form of 52, and divide it by 5

25 ÷ 5 = 5and 5 = 51

Now take that answer, 5, which is the standard form of 51, and divide it by 2.

5 ÷ 5 = 1

AND THEREFORE 1 =50Click to go back to where I left off