# Fractional Exponents

Also called **"Radicals"** or **"Rational Exponents"**

## Whole Number Exponents

First, let us look at whole number exponents:

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

In this example: **8 ^{2} = 8 × 8 = 64**

^{2}could be called "8 to the second power", "8 to the power 2" or simply "8 squared"

Another example: **5 ^{3} = 5 × 5 × 5 = 125**

## Fractional Exponents

But what if the exponent is a fraction?

An exponent of \frac{1}{2} is a An exponent of \frac{1}{3} is a An exponent of \frac{1}{4} is a And so on! |

## Why?

Let's see why in an example.

First, the Laws of Exponents tell us how to handle exponents when we multiply:

### Example: x^{2}x^{3} = (xx)(xxx) = xxxxx = x^{5}

Which shows that **x ^{2}x^{3} = x^{(2+3)} = x^{5}**

So let us try that with fractional exponents:

### Example: What is 9^{½} × 9^{½} ?

9^{½} × 9^{½} = 9^{(½+½)} = 9^{(1)} = 9

So 9^{½} times itself gives 9.

Now, what do we call a number that, when multiplied by itself, gives another number? The square root of that other number!

See:

√9 × √9 = 9

And:

9^{½} × 9^{½} = 9

So 9^{½} is the same as √9

## Try Another Fraction

Let us try that again, but with an exponent of one-quarter (1/4):

### Example:

16^{¼} × 16^{¼} × 16^{¼} × 16^{¼} = 16^{(¼+¼+¼+¼)} = 16^{(1)} = 16

So 16^{¼} used 4 times in a multiplication gives 16,

and so** 16 ^{¼} is a 4th root of 16**

## General Rule

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

x^{1/n} = The **n-**th Root of x

In other words:

**1/n**means to

**take the n-th root**:

x^{1/n} =
n√x

### Example: What is 27^{1/3} ?

Answer: 27^{1/3} =
3√27
= 3

## What About More Complicated Fractions?

What about a fractional exponent like 4^{3/2} ?

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

Let me explain.

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:

x^{m/n} =
x^{(m × 1/n)} =
(x^{m})^{1/n} =
n√x^{m}

The order does not matter, so it also works for **m/n = (1/n) × m**:

x^{m/n} =
x^{(1/n × m)} =
(x^{1/n})^{m} =
(n√x )^{m}

And we get this:

**m/n**means:

**m-th power**, then take the

**n-th root**: x

^{m/n}= n√x

^{m}

**n-th root**, then do the

**m-th power**: x

^{m/n}= (n√x )

^{m}

Some examples:

### Example: What is 4^{3/2} ?

4^{3/2} = 4^{3×(1/2)} = √(4^{3}) = √(4×4×4) = √(64) = **8**

or

4^{3/2} = 4^{(1/2)×3} = (√4)^{3} = (2)^{3} = **8**

Either way gets the same result.

### Example: What is 27^{4/3} ?

27^{4/3} = 27^{4×(1/3)}
= 3√27^{4}
= 3√531441
= **81**

or

27^{4/3} = 27^{(1/3)×4}
= (3√27 )^{4} = (3)^{4} = **81**

It was certainly easier the 2nd way!

## Now ... Play With The Graph!

See how *smoothly* the curve changes when you play with the fractions in this animation, this shows you that this idea of fractional exponents fits together nicely:

Things to try:

- Start with m=1 and n=1, then slowly increase n so that you can see 1/2, 1/3 and 1/4
- Then try m=2 and slide n up and down to see fractions like 2/3 etc
- Now try to make the exponent −1
- Lastly try increasing m, then reducing n, then
*reducing*m, then*increasing*n: the curve should go around and around