# ##\$ Fractional Exponents

In the world of math, fractional exponents are called “rational exponents” because a rational number is a number that can be written as a fraction.

However, a lot of people use the word “fractional” instead of “rational” because it’s easier to remember. You can use either word, but “rational” is the mathematically correct term.

Fractional (or rational) exponents are used to write roots and radicals in exponential form. Basic fractional exponents can be simplified like this…

The denominator of the fraction represents the type of root and the base of the power is the argument of the root.

## How to Simplify Fractional Exponents

1. Identify the type of root that corresponds with the denominator of the fraction.
2. Write the base of the power inside the root.
3. Evaluate the root, if possible.

Simplify $$1296^{\frac{1}{4}}$$

The denominator of the fractional exponent is a 4 and the base of the power is 1296. So, I will write 1296 inside of a 4th root.

$$1296^{\frac{1}{4}}= \sqrt[4]{1296}$$

There are no variables, so I can evaluate the root. 6 times itself four times is 1296, so the fourth root of 1296 is 6.

$$\sqrt[4]{1296}=6$$

## Inverse Operations

The operations of math build on each other.

Repeated counting becomes addition. Repeated addition becomes multiplication. And repeated multiplication becomes exponents.

The inverse of addition is subtraction and sometimes it’s helpful to write subtraction as adding a negative

The inverse of multiplication is division and sometimes it’s helpful to  write division as multiplying by a fraction.

The inverse of an exponent is a root. Fractional exponents allow us to write roots as exponents. This can be very helpful when you’re simplifying radicals or exponential expressions

## How to Write Roots as Fractional Exponents

You can also use fractional exponents to simplify roots.

1. Write the argument of the root as the base of the power.
2. Identify the type of root (square, cube, 4th, etc.)
3. Write a fractional exponent with a denominator that matches the type of root you identified.

Writing roots as fractional exponents can be helpful if you’re trying to find the root of a very large number. Once the root is written as a fractional exponent, you can type it into your calculator

It can also be very helpful to write roots as fractional exponents if there are variables involved. Once the root is written as a fractional exponent, you can use any of the exponent rules to simplify it.

Simplify $$\sqrt[3]{15625}$$

The argument of the root is 15625 and the root is a cube root. So, I’ll write a power with 15625 in the base and $$\frac{1}{3}$$ in the exponent.

$$\sqrt[3]{15625}=15625^{\frac{1}{3}}$$

Next, I can evaluate it by typing “15625 ^ ( 1 / 3 )” into my calculator and I get the answer 6.

$$\sqrt[3]{15625}=6$$

If you use a calculator to evaluate fractional exponents, make sure you type parentheses around the entire exponent so the calculator knows that there is a fraction is in the exponent.

### Correct

“15625 ^ ( 1 / 3 )” $$= 15625^{\frac{1}{3}}$$

### Incorrect

“15625 ^ 1 / 3” $$= \frac{15625^{1}}{3}$$

## Why It Works

Why did “they” choose fractional exponents to represent roots?

Why not choose a negative exponent? Or make up an entirely new kind of number to go in the exponent?

There are two reasons…

1. The way that roots and exponents interact with each other
2. The power rule of exponents

Roots and exponents are inverses. That means that they are exact opposites and each one cancels the effect of the other one.

For example, if you square a number and then square root it, you’re left with the original number. The same thing happens if you cube a number and then cube root it.

$(\sqrt{5^2})=5$

$(\sqrt[3]{4^3})=4$

$(\sqrt[n]{t^n})=t$

This also works in reverse. If you square root a number and then square it, you’ll end up with the original number.

The same thing happens with a cube root, a fourth root, a twelfth root, or any other root.

$(\sqrt{9})^{2}=9$

$(\sqrt[3]{8})^{3}=8$

So, what does this mean for fractional exponents?

When mathematicians were deciding what type of exponent they wanted to use to represent roots, they knew it had to meet both of these requirements…

$(\sqrt[n]{t^n})=t$

translates to…

$(t^{n})^{?}=t^1$

$(\sqrt[n]{x})^{n}=x$

translates to…

$(x^{?})^{n}=x^1$

Both of these expressions are examples of the power rule of exponents.

The power rule of exponents tells us that if we have a power raised to another exponent, we can multiply the two exponents to simplify the expression.

The simplified expressions (t and x) have exponents of 1. So, to satisfy the power rule of exponents…

$n(?)=1$

$?(n)=1\$

The only number that would satisfy both of those equations is $$\frac{1}{n}$$. So, that is why we use fractions to represent roots.