A rational exponent is a rational number that provides another method for writing roots. For example, an $n$th root of a number $b$ is a number $x$ such that $x^n = b$. If $b$ is a positive real number and $n$ is a positive integer, then there is exactly one positive real solution to $ x^n = b$. This solution is called the $n$th root of $b$ and is denoted $\sqrt[n]{b}$ or $b^\frac{1}{n}$. For example: $\sqrt{4}=4^\frac{1}{2}=2$.

There are also cases where the exponent is a fraction $\frac{m}{n}$, where $m$ is an integer and $n$ is a positive integer. In such cases, the exponent acts as both a whole number exponent and a root, or fraction exponent. In other words, the following holds true:

$\displaystyle{{b}^{\frac {m}{n}}= {({b}^{m})}^{\frac{1}{n}}=\sqrt[n]{{b}^{m}}}$

where $b$ is a real number and the rational exponent $\frac{m}{n}$ is a fraction in lowest terms.

The following rules hold true about the signs of roots and rational exponents. For a rational exponent $\frac{m}{n}$, where $\frac{m}{n}$ is in lowest terms:

• The root is positive if $m$ is even; for example, $(-27)^\frac{2}{3}=9$. 
• The root is negative for negative $b$ if $m$ and $n$ are odd; for example, $\displaystyle (-27)^\frac{1}{3}=-3$.  
• The root can be either sign if $b$ is positive and $n$ is even; for example, $64^\frac{1}{2}$has two roots: $8$ and $-8$. 

Note that since there is no real number $x$ such that $x^2 = -1$, the definition of $b^{\frac{m}{n}}$ when $b$ is negative and $n$ is even must involve the imaginary number $i$.

The following are rules for operations on numbers with rational exponents. 

Multiplying Numbers with Rational Exponents

The following holds true for any rational exponent:

$\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}$

For example, we can rewrite $\sqrt[3]{16}$ as a product: 

$\sqrt[3]{16}= \sqrt[3]{8} \cdot \sqrt[3]{2}$

Notice that $\sqrt[3]{8} = 2$, and therefore we have:

$\sqrt[3]{16} = 2\sqrt[3]{2}$

Dividing Numbers with Rational Exponents

The following holds true for any rational exponent:

$\sqrt[n]{\dfrac{a}{b}} = \dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}$

For example, we can rewrite $\sqrt{\frac{13}{9}}$ as a fraction with two radicals: 

$\sqrt{\dfrac{13}{9}} = \dfrac{\sqrt{13}}{\sqrt{9}}$

Notice that the denominator can be simplified further:

$\sqrt{9} = 3$

Therefore, the simplified form is:

$\sqrt{\dfrac{13}{9}} = \dfrac{\sqrt{13}}{3}$

Canceling Powers and Roots

In some cases, writing an exponent in its fraction form makes it easier to cancel powers and roots. Recall that $\sqrt[n]{{b}^{m}}= {({b}^{m})}^{\frac{1}{n}}= {b}^{\frac {m}{n}} $. We can use this rule to easily simplify a number that has both an exponent and a root. 

For example, consider $\sqrt[4]{5^8}$. This would take a long time to work out by hand, but consider how it can be rewritten using a rational exponent: 

$\sqrt[4]{5^8} = 5^{\frac{8}{4}}$

We can simplify the fraction in the exponent to 2, giving us $5^2=25$.

Example 1

Simplify the following expression:

$\sqrt{ \dfrac{3^8}{25}}$

This expression can be rewritten using the rule for dividing numbers with rational exponents:

$\sqrt{ \dfrac{3^8}{25}} = \dfrac{\sqrt{3^8}}{\sqrt{25}}$

Notice that the radical in the denominator is a perfect square and can therefore be rewritten as follows: 

$\sqrt{25} = 5$.

Now, notice that the numerator can be rewritten:

$\sqrt{3^8} = (3^8)^{\frac{1}{2}} = 3^{\frac{8}{2}}= 3^4$.

Therefore, the simplified form is:

$\sqrt{ \dfrac{3^8}{5}} = \dfrac{3^4}{5}$

Example 2

Simplify the following expression:

$\dfrac{\sqrt{7^5}}{\sqrt[4]{7^2}}$

First, rewrite the numerator and denominator in rational exponent form:

$\dfrac{7^{\frac{5}{2}}}{7^{\frac{2}{4}}}$

Notice that the exponent in the denominator can be simplified:

$\dfrac{7^{\frac{5}{2}}}{7^{\frac{1}{2}}}$

Recall the rule for dividing numbers with exponents, in which the exponents are subtracted. Applying the division rule, we have:

$\dfrac{7^{\frac{5}{2}}}{7^{\frac{1}{2}}} = 7^{\left(\frac{5}{2}-\frac{1}{2}\right)} = 7^{\frac{4}{2}} = 7^2$

Thus, the simplified form is simply $7^2 = 49$.