# Rational Exponents

How to simplify the numbers with rational exponents: definition, examples, and their solutions.

## Definition

The *n*th root means the exponent 1/*n*.

So ^{n}√*a*^{m} = *a*^{m/n}.*n*th Root

## Example 1: Rational Exponent of ^{3}√16

16 = 2^{4}

^{3}√2^{4} = 2^{4/3}

The radical is a cube root.

It's an odd root.

So you don't have to think about

the absolute value sign.*n*th Root

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

## Example 2: Radical Form of *a*^{7/9}

*a*^{7/9} = ^{9}√*a*^{7}

## Example 3: Simplify 4^{5/2}

4 = 2^{2}

So (given) = 2^{2⋅(5/2)}.

Power of a power

2⋅(5/2) = 5

2^{5} = 32

So (given) = 32.

## Example 4: Simplify 8^{-7/3}⋅8^{5/3}

8^{-7/3}⋅8^{5/3} = 8^{-7/3 + 5/3}

Product of powers

-7/3 + 5/3 = -2/3

8 = 2^{3}

So (given) = 2^{3⋅(-2/3)}.

3⋅(-2/3) = -2

2^{-2} = 1/2^{2}

Negative exponent

2^{2} = 4

So (given) = 1/4.

## Example 5: Simplify *x*^{-5/6}

*x*^{-5/6} = 1/*x*^{5/6}

Negative exponent

The denominator is *x*^{5/6},

which is an irrational number.

(*x*^{5/6} = ^{6}√*x*^{5})*x*^{5/6}⋅*x*^{1/6} = *x*

So, to rationalize the denominator,

multiply *x*^{1/6}/*x*^{1/6}.

Rationalizing a denominator

*x*^{5/6}⋅*x*^{1/6} = *x*^{5/6 + 1/6}

= *x*^{6/6}

Product of powers

*x*^{6/6} = *x*^{1}

= *x*

So (given) = *x*^{1/6}/*x*.

The given expression has a rational exponent.

So the answer should be using a rational exponent.

This means*x*^{1/6}/*x* is the answer

but ^{6}√*x*/*x* cannot be the answer.

## Example 6: Simplify (^{3}√*x*^{2}^{6}√*x*)/*x*

To combine these radicals,

write the radicals using rational exponents.^{3}√*x*^{2} = *x*^{2/3}^{6}√*x* = *x*^{1/6}

√*x* = *x*^{1/2}

The powers have the same base: *x*.

So combine the exponents.*x*^{1/6} is multiplied.

So write the exponent +1/6.

Product of powers*x*^{1/2} is divided.

So write the exponent -1/2.

(The sign is (-).)

Quotient of powers

2/3 = 4/6

1/2 = 3/6

So (given) = *x*^{4/6 + 1/6 - 3/6}.

4/6 + 1/6 - 3/6 = 2/6

So (given) = *x*^{2/6}.

2/6 = 1/3

The given expression is in radical form.

So the answer should be in radical form.

So change *x*^{1/3} in radical form.*x*^{1/3} = ^{3}√*x*

So ^{3}√*x* is the answer.

## Example 7: Simplify (*x*^{1/2} - 3)/(*x*^{1/2} + 1)

Given: (*x*^{1/2} - 3)/[*x*^{1/2} + 1]

[*x*^{1/2} + 1] means [√*x* + 1].

So the denominator [*x*^{1/2} + 1] is an irrational binomial.

The conjugate of [*x*^{1/2} + 1] is [*x*^{1/2} - 1].

So, to rationalize the denominator,

multiply (*x*^{1/2} - 1)/(*x*^{1/2} - 1).

Rationalizing a denominator

(*x*^{1/2} + 1)(*x*^{1/2} - 1) = (*x*^{1/2})^{2} - 1^{2}

Product of a sum and a difference (*a* + *b*)(*a* - *b*)

Think *x*^{1/2} as a variable

and expand the numerator.

(*x*^{1/2} - 3)(*x*^{1/2} - 1)

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

FOIL method

(*x*^{1/2})^{2} = *x*

-*x*^{1/2} - 3*x*^{1/2} = -4*x*^{1/2}

So (given) = (*x* - 4*x*^{1/2} + 3)/(*x* - 1).

The given expression has a rational exponent.

So the answer should be using a rational exponent.

So this is the answer.