In American mathematics courses, the knowledge of exponents is critical, particularly on standardized tests such as the SAT, ACT, and AP tests. The exponent of a number tells us how many times the base is raised to itself. When the exponent is a fraction, it is referred to as a fractional exponent—a versatile tool that unifies powers and roots into a single notation. In this in-depth guide, we will cover all U.S. students need to know regarding fractional exponents in order to pass their exams. You will learn how to define fractional exponents, learn the rules that apply to them, and be able to simplify, multiply, and divide them. We will also review helpful tips and practice problems to lock these into place so that you're completely ready for any math test that includes fractional exponents.

What are Fractional Exponents?

A fractional exponent is a way for expressing powers and roots together. For example, the following are equivalent. For example, the following are equivalent.

In any general exponential expression of the form \(a^{b}\), where \(a\) is the base and \(b\) is the exponent. When \(b\) is given in the fractional form, it is known as a fractional exponent. Few examples of fractional exponents are \(2^{\frac{1}{5}}\), \(2^{\frac{3}{5}}\), \(2^{\frac{2}{3}}\).

The general form of a fractional exponent is \(x^{\frac{m}{n}}\), where \(x\) is the base and \(\frac{m}{n}\) is the exponent. Look at the figure given below to understand how fractional exponents are represented.

Some examples of fractional exponents that are widely used are given below:

Fractional Exponents Laws

There are certain laws to be followed that help to multiply or divide numbers with fractional exponents easily are given below:

• \(a^{\frac{1}{m}}\times a^{\frac{1}{n}}=a^{(\frac{1}{m}+\frac{1}{n})}\).
• \(a^{\frac{1}{m}}\div a^{\frac{1}{n}}=a^{(\frac{1}{m}-\frac{1}{n})}\).
• \(a^{\frac{1}{m}}\times b^{\frac{1}{m}}=(ab)^{\frac{1}{m}}\).
• \(a^{\frac{1}{m}}\div b^{\frac{1}{m}}=(a\div b)^{\frac{1}{m}}\).
• \(a^{-\frac{m}{n}}=\left(\frac{1}{a}\right)^{\frac{m}{n}}\).

Simplifying Fractional Exponents

Simplifying fractional exponents can be understood in two ways (multiplication and division). It involves reducing the expression or the exponent to a reduced form which is easy to understand. Let us understand the simplification of fractional exponents with the help of some examples.

Example 1: Simplify \(125^{\frac{1}{3}}\).

Solution: \(125^{\frac{1}{3}}=\sqrt[3]{125}\)

\(\Rightarrow\) \(125^{\frac{1}{3}}=[(5)^{3}]^{\frac{1}{3}}\)

\(\Rightarrow\) \(125^{\frac{1}{3}}=(5)^{1}\)

\(\Rightarrow\) \(125^{\frac{1}{3}}=5\)

Therefore, \(125^{\frac{1}{3}} = 5\).

Example 2: Find \(4^{\frac{3}{2}}\).

Solution: \(4^{\frac{3}{2}}=4^{3\times \frac{1}{2}}\)

\(\Rightarrow\) \(4^{\frac{3}{2}}=\sqrt{4^{3}}\)

\(\Rightarrow\) \(4^{\frac{3}{2}}=\sqrt{4\times 4\times 4}\)

\(\Rightarrow\) \(4^{\frac{3}{2}}=\sqrt{64}\)

\(\Rightarrow\) \(4^{\frac{3}{2}}=8\)

Alternate solution:

\(4^{\frac{3}{2}}=4^{(\frac{1}{2})\times 3}\)

\(\Rightarrow\) \(4^{\frac{3}{2}}=(\sqrt{4})^{3}\)

\(\Rightarrow\) \(4^{\frac{3}{2}}=(2)^{3}\)

\(\Rightarrow\) \(4^{\frac{3}{2}}=8\).

Therefore, \(4^{\frac{3}{2}}=8\).

Multiplying Fractional Exponents

The multiplication of fractional exponents can be classified into two types:

1. Multiplication of fractional exponents with different powers but same bases: Multiplying terms having the same bases and with different fractional exponents is equal to adding together the exponents. The general rule for multiplying exponents with the same bases and different fractional exponents is \(a^{\frac{1}{m}}\times a^{\frac{1}{n}}=a^{(\frac{1}{m}+\frac{1}{n})}\).

For example: Workout \(2^{\frac{2}{3}}\times 2^{\frac{3}{4}}\).

Solution: \(2^{\frac{2}{3}}\times 2^{\frac{3}{4}}=2^{(\frac{2}{3}+\frac{3}{4})}\)

\(\Rightarrow\) \(2^{\frac{2}{3}}\times 2^{\frac{3}{4}}=2^{\frac{17}{12}}\)

Therefore, \(2^{\frac{2}{3}}\times 2^{\frac{3}{4}}=2^{\frac{17}{12}}\).

2. Multiplication of fractional exponents with different bases but same powers: Multiplying terms having the different bases and with same fractional exponents is equal to multiplying the bases together. The general rule for multiplying exponents with different bases and same fractional exponents is \(a^{\frac{1}{m}}\times b^{\frac{1}{m}}=(ab)^{\frac{1}{m}}\).

For example: Workout \(4^{\frac{1}{3}}\times 2^{\frac{1}{3}}\).

Solution: \(4^{\frac{1}{3}}\times 2^{\frac{1}{3}}=(4\times 2)^{\frac{1}{3}}\)

\(\Rightarrow\) \(4^{\frac{1}{3}}\times 2^{\frac{1}{3}}=(8)^{\frac{1}{3}}\)

Therefore, \(4^{\frac{1}{3}}\times 2^{\frac{1}{3}}=(8)^{\frac{1}{3}}\).

Dividing Fractional Exponents

The division of fractional exponents can be classified into two types:

1. Division of fractional exponents with different powers but the same bases: The general rule for dividing exponents with the same bases and different fractional exponents is \(a^{\frac{1}{m}}\div a^{\frac{1}{n}}=a^{(\frac{1}{m}-\frac{1}{n})}\).

For example: Workout \(2^{\frac{3}{4}}\div 2^{\frac{1}{2}}\).

Solution: \(2^{\frac{3}{4}}\div 2^{\frac{1}{2}}=2^{(\frac{3}{4}-\frac{1}{2})}\)

\(\Rightarrow\) \(2^{\frac{3}{4}}\div 2^{\frac{1}{2}}=2^{\frac{1}{4}}\)

Therefore, \(2^{\frac{3}{4}}\div 2^{\frac{1}{2}}=2^{\frac{1}{4}}\).

2. Division of fractional exponents with the same powers but different bases: The general rule for dividing exponents with different bases and same fractional exponents is \(a^{\frac{1}{m}}\div b^{\frac{1}{m}}=(a\div b)^{\frac{1}{m}}\).

For example: Workout \(4^{\frac{1}{3}}\div 2^{\frac{1}{3}}\).

Solution: \(4^{\frac{1}{3}}\div 2^{\frac{1}{3}}=(4\div 2)^{\frac{1}{3}}\)

\(\Rightarrow\) \(4^{\frac{1}{3}}\div 2^{\frac{1}{3}}=(2)^{\frac{1}{3}}\)

Therefore, \(4^{\frac{1}{3}}\div 2^{\frac{1}{3}}=(2)^{\frac{1}{3}}\).

Negative Fractional Exponents

Negative fractional exponents are the same as rational exponents. To solve negative exponents, we have to apply exponents rules that say \(a^{-m}=\frac{1}{a^{m}}\). The general rule for negative fractional exponents is \(a^{-\frac{m}{n}}=\left(\frac{1}{a}\right)^{\frac{m}{n}}\).

For example: Simplify \(9^{-\frac{1}{2}}\).

Solution: \(9^{-\frac{1}{2}}=\left(\frac{1}{9}\right)^{\frac{1}{2}}\)

\(\Rightarrow\) \(9^{-\frac{1}{2}}=\left[\left(\frac{1}{3}\right)^{2}\right]^{\frac{1}{2}}\)

\(\Rightarrow\) \(9^{-\frac{1}{2}}=\left(\frac{1}{3}\right)^{1}\)

\(\Rightarrow\) \(9^{-\frac{1}{2}}=\frac{1}{3}\)

Therefore, \(9^{-\frac{1}{2}}=\frac{1}{3}\).

Radical Exponents Using Fractional Exponents

Square roots are most often written using a radical sign, like this, \(\sqrt{4}\). But there is another way to represent them. You can use a fractional exponent instead of a radical. A rational exponent (or fractional exponent) is an exponent that is a fraction. For example, \(\sqrt{4}\) can be written as \(4^{\frac{1}{2}}\).

Radical exponents and fractional exponents are alternate ways of expressing the same thing. In the table below we show equivalent ways to express radical exponents and fractional exponents.

Fractional Exponents Solved Examples

1. Simplify the expression, \(8^{\frac{1}{3}}\times 8^{\frac{1}{3}}\).

Solution: \(8^{\frac{1}{3}}\times 8^{\frac{1}{3}}=8^{(\frac{1}{3}+\frac{1}{3})}\)

\(\Rightarrow\) \(8^{\frac{1}{3}}\times 8^{\frac{1}{3}}=8^{\frac{2}{3}}\)

\(\Rightarrow\) \(8^{\frac{1}{3}}\times 8^{\frac{1}{3}}=[(2)^{3}]^{\frac{2}{3}}\)

\(\Rightarrow\) \(8^{\frac{1}{3}}\times 8^{\frac{1}{3}}=(2)^{2}\)

\(\Rightarrow\) \(8^{\frac{1}{3}}\times 8^{\frac{1}{3}}=4\)

Therefore, \(8^{\frac{1}{3}}\times 8^{\frac{1}{3}}=4\).

2. Simplify the expression, \(16^{\frac{1}{2}}\div 16^{\frac{1}{4}}\).

Solution: \(16^{\frac{1}{2}}\div 16^{\frac{1}{4}}=16^{(\frac{1}{2}-\frac{1}{4})}\)

\(\Rightarrow\) \(16^{\frac{1}{2}}\div 16^{\frac{1}{4}}=16^{(\frac{2}{4}-\frac{1}{4})}\)

\(\Rightarrow\) \(16^{\frac{1}{2}}\div 16^{\frac{1}{4}}=16^{\frac{1}{4}}\)

\(\Rightarrow\) \(16^{\frac{1}{2}}\div 16^{\frac{1}{4}}=[(2)^{4}]^{\frac{1}{4}}\)

\(\Rightarrow\) \(16^{\frac{1}{2}}\div 16^{\frac{1}{4}}=(2)^{1}\)

\(\Rightarrow\) \(16^{\frac{1}{2}}\div 16^{\frac{1}{4}}=2\)

Therefore, \(16^{\frac{1}{2}}\div 16^{\frac{1}{4}}=2\).

We hope this article has given useful insights on how to better your knowledge of fractional exponents and improve your understanding to prepare for your exams. For U.S. students working on competitive exams such as the SAT, ACT, GMAT, and GRE, a good grip on these concepts is crucial. Stay tuned with the Testbook App for further updates on important topics in mathematics and other relevant topics. Do not forget to visit the test series available to judge your knowledge and enhance your confidence level before approaching these exams.