## 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.

6^{2} = 6 × 6 = 36

4^{3} = 4 × 4 × 4 = 64

In words:

6^{2} 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:

x^{1/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:

What is 8^{1/3} ?

Answer:

8^{1/3} = ^{3}√8 = 2

## More Complicated Fractions

What about a fractional exponent like 9^{3/2} ?

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:

- a whole number part (m) , and
- 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 | 9^{3/2} = ^{2}√(9^{3}) = ^{2}√729 = 27 | |

Or Take the n-th root and then do the m-th power | 9^{3/2} = (^{2}√9)^{3} = 3^{3} = 27 |

If they give you x^{3/6}, then x had better not be negative, because x^{3} would still be negative, and you would be trying to take the sixth root of a negative number.

If they give you x^{4/6}, 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 x^{4/5}, then you don’t need to care whether x is positive or negative, because a fifth root doesn’t have any problem with negatives.