Rational Exponents

• When a base is raised to a rational exponent of the form 1/n we use the following definition:

The denominator of the rational exponent …

1nnb b

… becomes the index of the radical ...

…and the base of the exponential …

… becomes the radicand of the radical.

• Example 1

Write each of the following using radical notation. Simplify the radical if possible:

1 45 4 5

1 3xyz 3 xyz

1 216 16 4

• Example 2

Write each of the following in rational exponent form:

5 2 1 52

4 xyz 1 4xyz

mn b

• When a base is raised to a rational exponent of the form m/n we use the following definition:

m n n mb b

The numerator m becomes the … exponent.

mn b

• When a base is raised to a rational exponent of the form m/n we use the following definition:

m n n mb b

The numerator m becomes the … exponent.

The denominator n becomes the … index.

mn b

• When a base is raised to a rational exponent of the form m/n we use the following definition:

m n n mb b

The numerator m becomes the … exponent.

The denominator n becomes the … index.

The base b becomes the … radicand.

• Example 3

3 45

34 5

4 125

Write the following using radical notation. Simplify the radical if possible:

34 5or

2 38

3 28

3 64

23 8or

4

22

4

Whenever the radical can be simplified easily, the second form is the one to use.

• Example 4

• Example 5

Write the following in rational exponent form:

34 xyz 3 4xyz

• When a base is raised to a negative rational exponent we use the following definition:

1m nm n

bb

• When a rational base is raised to a negative rational exponent we use the following:

m n m na b

b a

Take the reciprocal of the rational expression and make the exponent positive.

• Example 6

Write the following using positive exponents. Simplify if possible:

2 382 3

1

8

23

1

8

2

1

2

1

4

• Example 7

3 5xyz

3 5

1

xyz

• Example 8

4 38

27

4 327

8

Take the reciprocal of the rational expression and make the exponent positive.

• Example 8

4 38

27

4 327

8

4

327

8

43

2

81

16