# Fractional Exponents

A fractional exponent is an alternate notation for expressing powers and roots together. Fractional Exponents are also known as Radicals or Rational Exponents.

Table of Contents

## Whole Number Exponents

First, look at whole number exponents:

The exponent of a number says how many times to use the number in a multiplication.

Examples

62 = 6 × 6 = 36

43 = 4 × 4 × 4 = 64

In words:

62 could be called “6 to the second power”, or “6 to the power 2” or simply “6 squared”

## Fractional Exponents

Remember that when aa is a positive real number, both of these equations are true:

If the exponent is a fraction?

When you have a fractional exponent, the numerator is the power and the denominator is the root. In the variable. So:

Where:

• x is a real number
• a and b are positive real numbers
• a is the power
• b is the root

 Exponent of Fraction Example An exponent of ½ is square root  An exponent of ⅓ is cube root  An exponent of ¼ is 4th root  And so on! etc

## General Rule

It worked for ½, it worked with ¼, in fact it works generally:

x1/n = The n-th Root of x

So we can come up with this

A fractional exponent like 1/n means to take the n-th root:

Example

What is 81/3 ?

Answer:

81/3 = 3√8 = 2

## More Complicated Fractions

What about a fractional exponent like 93/2 ?

That is really saying to do a cube (3) and a square root (1/2), in any order.

A fraction (like m/n) can be broken into two parts:

1. a whole number part (m) , and
2. a fraction (1/n) part

So, because m/n = m × (1/n) we can do this:

And we get this:

A fractional exponent like m/n means:

 In Symbols Example Do the m-th power, then take the n-th root  93/2 = 2√(93) = 2√729 = 27 Or Take the n-th root and then do the m-th power  93/2 = (2√9)3 = 33 = 27

Note

If they give you x3/6, then x had better not be negative, because x3 would still be negative, and you would be trying to take the sixth root of a negative number.

If they give you x4/6, then a negative x becomes positive (because of the fourth power) and then it is sixth-rooted, so by reducing the fractional power it becomes |x|2/3.

But, if they give you something like x4/5, then you don’t need to care whether x is positive or negative, because a fifth root doesn’t have any problem with negatives.

## Learn More

Exponents

Negative Exponents

Simplifying Square Roots

Cube Numbers and Cube Roots

Algebra Index