# Rational Exponent

How to solve and use a rational exponent: formula, 5 examples, and their solutions.

## Formula

### Formula

^{n}√a^{m} = a^{m/n}

## Example 1

### Example

### Solution

16 = 2^{4}

Power

^{3}√2^{4} = 2^{4/3}

So 2^{4/3} is the answer.

## Example 2

### Example

### Solution

a^{7/9} = ^{9}√a^{7}

So ^{9}√a^{7} is the answer.

## Example 3

### Example

### Solution

4^{5/2}

= (2^{2})^{5/2}

= 2^{2⋅[5/2]}

Power of a Power

2⋅[5/2] = 5

2^{5} = 32

So 32 is the answer.

## Example 4

### Example

### Solution

3^{-7/2}⋅3^{5/2} = 3^{-7/2 + 5/2}

Product of Powers

-7/2 + 5/2 = -2/2

-2/2 = -1

3^{-1} = 1/3

Negative Exponent

So 1/3 is the answer.

## Example 5

### Example

### Solution

Change each radical

to a rational exponent power.

^{3}√x^{2} = x^{2/3}

^{6}√x

= ^{6}√x^{1}

= x^{1/6}

√x

= ^{2}√x^{1}

= x^{1/2}

So

[^{3}√x^{2}⋅^{6}√x]/√x

= [x^{2/3}⋅x^{1/6}]/x^{1/2}.

[x^{2/3}⋅x^{1/6}]/x^{1/2}

= x^{2/3 + 1/6 - 1/2}

Change the denominators to 6.

[2/3]⋅[2/2] = 4/6

-[1/2]⋅[3/3] = -3/6

4/6 + 1/6 - 3/6

= 5/6 - 3/6

= 2/6

2/6 = 1/3

x^{1/3}

= ^{3}√x^{1}

= ^{3}√x

So ^{3}√x is the answer.