Working with Negative Exponents

When working with negative exponents, move the variable and change the exponent

to positive.

When working with negative exponents, move the variable and change the exponent

to positive.

3

1

2x

31

2

x

3

2

x

When working with negative exponents, move the variable and change the exponent

to positive.

4

7

y

4

1

7y

When working with negative exponents, move the variable and change the exponent

to positive.

9

2

3

4

x

y

2

9

3

4

y

x

When working with negative exponents, move the variable and change the exponent

to positive.

7 5

3

2

5

x y

xz

3

7 5

2

5

z

xx y

3

8 5

2

5

z

x y

Simplifying Square Roots

21 122 423 924 1625 25

26 3627 4928 6429 81210 100

211 121212 144213 169214 196215 225

Simplify the Square Root

25 5

Simplify the Square Root

81 9

Simplify the Square Root

144 12

25 5

81 9

144 12

Perfect Squares

54 ?

72 ?

80 ?

Not Perfect Squares

Simplify the Square Root

1, 4, 9

16, 25, 36

49, 64, 81

100, 121, 144

169, 196, 225

What’s the biggest number here What’s the biggest number here that will go into 54 evenly?that will go into 54 evenly?

9 x 69 x 6

54

Simplify the Square Root

54 9 6 3 6 3 6

Simplify the Square Root

1, 4, 9

16, 25, 36

49, 64, 81

100, 121, 144

169, 196, 225

What’s the biggest number here What’s the biggest number here that will go into 72 evenly?that will go into 72 evenly?

36 x 236 x 2

72

Simplify the Square Root

72 36 2 6 2 6 2

Simplify the Square Root

1, 4, 9

16, 25, 36

49, 64, 81

100, 121, 144

169, 196, 225

What’s the biggest number here What’s the biggest number here that will go into 80 evenly?that will go into 80 evenly?

16 x 516 x 5

80

Simplify the Square Root

80 16 5 4 5 4 5

The square root of a negative number does not exist.

So, we say it is an imaginary number and we use the variable “i”.

The square root of a negative number does not exist.

So, we say it is an imaginary number and we use the variable “i”.

16 4i

The square root of a negative number does not exist.

So, we say it is an imaginary number and we use the variable “i”.

100 10i

Simplify the Square Root

1, 4, 9

16, 25, 36

49, 64, 81

100, 121, 144

169, 196, 225

What’s the biggest number here What’s the biggest number here that will go into 8 evenly?that will go into 8 evenly?

4 x 24 x 2

8 8i

Simplify the Square Root

8 8i4 2i 2 2i

2 2i

Multiplying Square Roots

Multiply the Square Roots

2 8 16 4

Multiply the Square Roots

5 20 100

10

Multiply the Square Roots

3 12 36

6

Fractions as exponents

3 54 Square Root

Cube Root

Fourth Root

Fifth Root

4 2

What do you multiply by itself to get 4?

3 27 3

What do you multiply by itself 3 times to get 27?

3 3 3 9 327

4 16 2

What do you multiply by itself 4 times to get 16?2 2 2 2 4 2 2

8 216

1

2as exponent

means 1

3as exponent

means3

1

4as exponent

means4

1

5as exponent

means5

1225 25 5

138 3 8 2

15x 5 x

1

2as exponent

means 1

3as exponent

means3

1

4as exponent

means4

1

5as exponent

means5

Numerator is 1

If the numerator is not 1, it becomes the exponent underneath the radical.

23x Outside Radical

Under Radical

If the numerator is not 1, it becomes the exponent underneath the radical.

23 x2

3x

If the numerator is not 1, it becomes the exponent underneath the radical.

79z 79 z

If the numerator is not 1, it becomes the exponent underneath the radical.

45y

45 y

If there are 2 exponents, multiply them.

3

1 42x

38x

If there are 2 exponents, multiply them.

2

4 35y

815y