Radicals and rational exponents

So far, you’ve worked with integer exponents. In this chapter, you’ll explore fractional exponents and how to write them as radicals. A fractional exponent is an exponent that isn’t a whole number, like $x^{\frac{1}{2}}$.

Denominator of fractional exponents

Square root

A fractional exponent “undoes” an integer exponent. For example:

$$4^2=16$$

And:

$$4^{\frac{1}{2}}=2$$

When the exponent is $\frac{2}{1}$, you square the number: $4\cdot 4=16$. When the exponent is $\frac{1}{2}$, you’re asking for the number that gives $4$ when squared. That number is $2$, because $2\cdot 2=4$.

This is the square root. You may already know that a square root undoes squaring. The key idea here is that a root (also called a radical) undoes an exponent in general.

Cube root

A cubed number is a number raised to the power of $3$. The opposite operation corresponds to raising to the power of $\frac{1}{3}$. Like the square root, the fractional exponent $\frac{1}{3}$ can be written using a radical and is called the cube root.

Compare the square root and cube root notations:

$$x^{\frac{1}{2}}=\sqrt{x}$$

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

Notice that the cube root has a $3$ written on the radical, but the square root does not show a $2$. This is the standard convention:

• If there is no number on the radical, it means square root (index $2$).
• If there is a number on the radical, that number is the denominator of the fractional exponent.

The number written on the radical is called the index.

Numerator of fractional exponents

Fractional exponents won’t always have a $1$ in the numerator. When there is an integer in the numerator, that number becomes an exponent inside the radical.

For example:

$$x^{\frac{2}{3}}=\sqrt[3]{x^2}$$

Here, the $2$ goes inside the radical as the exponent on $x$.

At this point, you have the rules you need to convert between fractional exponents and radicals. You’ll sometimes be asked to do this because fractional exponents can be harder to read in an expression. In many cases, writing the radical form is clearer.

Conversion practice

You need to be comfortable converting from a fractional exponent to a radical and from a radical to a fractional exponent. These are core concepts that are frequently tested, so let’s work through several examples.

What is $\sqrt[4]{x^2}$ as an exponent?

The numerator of the fractional exponent is the exponent inside the radical, $2$. The denominator is the index on the radical, $4$.

So the exponent form is $x^{\frac{2}{4}}$, which simplifies to $x^{\frac{1}{2}}$.

What is the exponent form of $\sqrt[3]{x^3}$?

What is $x^{\frac{5}{2}}$ expressed as a radical?

What is $x^{-\frac{2}{3}}$ expressed as a radical?

Rationalizing a fraction with a radical in the denominator

Consider this expression:

$$\frac{1}{\sqrt{5}}$$

This fraction has a radical in the denominator. In this course, you’ll rewrite such expressions so the denominator is a rational number (no square roots, cube roots, etc.). This process is called rationalizing the denominator.

The method depends on the type of radical. In this example, multiply by $\frac{\sqrt{5}}{\sqrt{5}}$, which equals $1$:

$$\frac{1}{\sqrt{5}}\times \frac{\sqrt{5}}{\sqrt{5}}=\frac{\sqrt{5}}{5}$$

Now the denominator is rational.