Fractional Exponents

Fractional Exponents

Radical Form Values with a fractional exponent can be written in radical form as follows:

Exponential Form

Radical Form radical sign

a

1 n

index

=

n

a radicand

Examples:

1 3

x =

3

x

x =x

1 2

Note: When a radical sign has no index value, the value is assumed to be 2.

Roots A fractional exponent or radical sign represents a root. Roots are the inverse of exponents.

Examples:

36 =

6×6 =

62 = 6

The second (or square) root of 36 is 6.

3

27 =

3

3×3×3 =

3

The third (or cube) root of 27 is 3.

4

8 x

=

4

2 4 (x )

The fourth root of x8 is x2.

=

2 x

33 = 3

Fractional Exponents Exponent rules still apply to fractional exponents.

Applying the Power of a Power rule:

a

m n

Example:

= (a

m

1 n

1 n

m

) = (a ) 16

3 4 =

(

4

=

n

m a

=( a ) 16

OR

n

m

3

4 4 2

) =(

)

3

=

23

=

8

Evaluate the following expression:

625

a. b. c. d. e.

172.5 –172.5 3,125 –3,125 none of the above

– 5 4

Evaluate the following expression:

625

– 5 4

Simplify the following expression. Express your solution in radical form. 3 (49x3y2) 2

Simplify the following expression. Express your answer as a single fraction.

8a–3 +

– 1 (64a6) 2