Fractional Exponents - Examples and Practice Problems

Fractional exponent exercises can be solved using the fractional exponent rule. This rule indicates the relationship between powers and radicals. The denominator of a fractional exponent is written as a radical of the expression and the numerator is written as the exponent.

Here, we will see a brief summary of fractional exponents in algebraic expressions. We will also look at various fractional exponent problems to learn how to solve these types of problems.

ALGEBRA

fractional exponents rules and exercises

Relevant for…

Learning to solve fractional exponent problems.

See examples

Contents

ALGEBRA

Relevant for…

Learning to solve fractional exponent problems.

See examples

Summary of fractional exponents

A fractional exponent is a technique for expressing powers and roots together. The general form of a fractional exponent is:

transform from radical to fractional exponent

We can define the following terms:

Radicand: The radicand is the expression under the sign √. In the expression above, the radicand is ${{b}^m}$ .
Index: The index or also known as the order of the radical, is the number that indicates which root is being applied. In the expression above, the index is n.
Base: The base is the number to which the root or power applies. In this case, the base is b.
Power: Power indicates repeated multiplication of the base by itself. In the expression above, the power is m.

Fractional exponents – Examples with answers

Each of the following examples has a detailed solution. The solution can be used to master the process of solving exercises with fractional exponents. Try to solve the exercises yourself before looking at the solution.

EXAMPLE 1

Simplify the expression ${{3}^{\frac{3}{2}}}$ .

Solution

The fractional exponent rule tells us that ${{b}^{\frac{m}{n}}}=\sqrt[n]{{{b}^m}}$ . Therefore, we write 3 to the power of 3 and then we take the square root of this:

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

Now, we simplify the expression by applying the exponent of 3:

$\sqrt[2]{{{3}^3}}}=\sqrt[2]{81}$

We can simplify again by recognizing that the square root of 81 is 9:

$\sqrt[2]{81}=9$

EXAMPLE 2

Simplify the expression ${{4}^{\frac{2}{3}}}$ .

Solution

Now, we have to write 4 raised to the power of 2 and we have to take the cube root of that expression:

${{4}^{\frac{2}{3}}}=\sqrt[3]{{{4}^2}}}$

We simplify by applying the exponent:

$\sqrt[3]{{{4}^2}}}=\sqrt[3]{16}$

We can simplify by rewriting 16 as 8 × 2:

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

The cube root of 8 is 2, so we have:

$\sqrt[3]{8\times 2}=2\sqrt[3]{2}$

EXAMPLE 3

Simplify the expression ${{-2}^{\frac{4}{3}}}{{x}^{\frac{2}{3}}}$ .

Solution

Here we have a number and a variable. Raise -2 to the fourth power and take its cube root and square x and take its cube root:

${{-2}^{\frac{4}{3}}}{{x}^{\frac{2}{3}}}=\sqrt[3]{{{(-2)}^4}}}\sqrt[3]{{{x}^2}}}$

We can apply the exponent to -2 to simplify:

$\sqrt[3]{{{(-2)}^4}}}\sqrt[3]{{{x}^2}}}=\sqrt[3]{16}\sqrt[3]{{{x}^2}}}$

Similar to the previous problem, we can simplify by rewriting 16 as 8 × 2:

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

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

Therefore, we have:

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

Now, we can combine the cube roots to simplify:

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

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EXAMPLE 4

Simplify the expression ${{6}^{\frac{3}{2}}}{{x}^{\frac{5}{2}}}$ .

Solution

We write 6 cubed and take its square root. We write to the x raised to the fifth and take its square root:

${{6}^{\frac{3}{2}}}{{x}^{\frac{5}{2}}}=\sqrt{{{6}^3}}}\sqrt{{{x}^5}}}$

We simplify 6 cubed:

$\sqrt{{{6}^3}}}\sqrt{{{x}^5}}}=\sqrt{216}\sqrt{{{x}^5}}}$

It is possible to simplify by writing to 216 as 36 × 6:

$\sqrt{216}=\sqrt{36\times 6}$

$=6\sqrt{6}$

Therefore, we have:

$\sqrt{216}\sqrt{{{x}^5}}}=6\sqrt{6}\sqrt{{{x}^5}}}$

Combining the square roots, we have:

$6\sqrt{6}\sqrt{{{x}^5}}}=6\sqrt{6{{x}^5}}}$

EXAMPLE 5

Simplify the expression ${{4}^{-\frac{3}{2}}}{{x}^{\frac{1}{2}}}$ .

Solution

In this case, we have a negative exponent. Remember that a negative exponent can be transformed to positive by taking the reciprocal of the base. Therefore, we have:

${{4}^{-\frac{3}{2}}}{{x}^{\frac{1}{2}}}=\frac{{{x}^{\frac{1}{2}}}}{{{4}^{\frac{3}{2}}}}$

Now, we cube 4 and take its square root and take the square root of the x:

$\frac{{{x}^{\frac{1}{2}}}}{{{4}^{\frac{3}{2}}}}=\frac{\sqrt{x}}{\sqrt{{{4}^3}}}$

We can apply the exponent to 4 to simplify:

$\frac{\sqrt{x}}{\sqrt{{{4}^3}}}=\frac{\sqrt{x}}{\sqrt{64}}$

Now, we can take the square root of 64:

$\frac{\sqrt{x}}{\sqrt{64}}=\frac{\sqrt{x}}{8}$

EXAMPLE 6

Simplify the expression ${{12}^{-\frac{2}{3}}}{{x}^{\frac{3}{5}}}$ .

Solution

We start transforming the exponent to positive by taking the reciprocal of the base. Therefore, we have:

${{12}^{-\frac{2}{3}}}{{x}^{\frac{3}{5}}}=\frac{{{x}^{\frac{3}{5}}}}{{{12}^{\frac{2}{3}}}}$

Now, we square 12 and take its cube root. We cube x and take its fifth root:

$\frac{{{x}^{\frac{3}{5}}}}{{{12}^{\frac{2}{3}}}}=\frac{\sqrt[5]{{{x}^3}}}{\sqrt[3]{{{12}^2}}}$

We apply the exponent to 12:

$\frac{\sqrt[5]{{{x}^3}}}{\sqrt[3]{{{12}^2}}}=\frac{\sqrt[5]{{{x}^3}}}{\sqrt[3]{144}}$

We can write 144 as 8×18 and take the cube root of 8:

$\frac{\sqrt[5]{{{x}^3}}}{\sqrt[3]{144}}=\frac{\sqrt[5]{{{x}^3}}}{\sqrt[3]{8\times 18}}$

$=\frac{\sqrt[5]{{{x}^3}}}{2\sqrt[3]{18}}$

Fractional exponents – Practice problems

Test your skills and your knowledge of fractional exponents with the following problems. Solve the problems and select an answer. Check it to see if you selected the correct answer. Use the solved examples above in case you need help.