# 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

Relevant for

Learning to solve fractional exponent problems.

See examples

##### 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:

We can define the following terms:

• Radicand: The radicand is the expression under the sign √. In the expression above, the radicand is $latex {{b}^m}$.
• 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.

## Radical form to fractional exponent

To transform from radical form to fractional exponent, we have to use the fractional exponent rule inversely.

We can form a fractional exponent where the numerator is the exponent to which the base is raised and the denominator is the index of the radical. That is, we use the following relationship:

## Fractional exponents – Examples with answers

### EXAMPLE 1

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

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

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

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

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

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

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

### EXAMPLE 2

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

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

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

We simplify by applying the exponent:

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

We can simplify by rewriting 16 as 8 × 2:

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

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

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

### EXAMPLE 3

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

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:

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

We can apply the exponent to -2 to simplify:

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

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

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

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

Therefore, we have:

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

Now, we can combine the cube roots to simplify:

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

### EXAMPLE 4

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

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

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

We simplify 6 cubed:

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

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

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

$latex =6\sqrt{6}$

Therefore, we have:

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

Combining the square roots, we have:

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

### EXAMPLE 5

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

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:

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

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

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

We can apply the exponent to 4 to simplify:

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

Now, we can take the square root of 64:

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

### EXAMPLE 6

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

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

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

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

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

We apply the exponent to 12:

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

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

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

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

### EXAMPLE 7

Simplify the expression $latex {{x}^{{\frac{1}{2}}}}{{y}^{{\frac{2}{3}}}}$.

We simply apply the rule of fractional exponents to form radicals:

$latex {{x}^{{\frac{1}{2}}}}{{y}^{{\frac{2}{3}}}}=\sqrt{x}~\sqrt[3]{{{{y}^{2}}}}$

### EXAMPLE 8

Simplify the expression $latex {{81}^{{\frac{1}{4}}}}{{x}^{{\frac{1}{2}}}}$

Again, we just have to apply the rule of fractional exponents to form radicals and then we simplify:

$latex {{81}^{{\frac{1}{4}}}}{{x}^{{\frac{3}{2}}}}=\sqrt[4]{{81}}~\sqrt{{{{x}^{3}}}}$

$latex =3~\sqrt{{{{x}^{3}}}}$

### EXAMPLE 9

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

Here, we have negative exponents, so we start by transforming negative exponents to positive using the negative exponents rule:

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

Now, we use the fractional exponent rule and simplify:

$latex =\frac{1}{{\sqrt{4}~\sqrt{x}}}$

$latex =\frac{1}{{2~\sqrt{x}}}$

### EXAMPLE 10

Simplify the expression $$\frac{{{{{16}}^{{-\frac{1}{2}}}}~{{y}^{{-\frac{1}{3}}}}}}{{{{x}^{{-\frac{1}{2}}}}~}}$$

We have negative exponents, so we start with the negative exponents rule:

$$\frac{{{{{16}}^{{-\frac{1}{2}}}}~{{y}^{{-\frac{1}{3}}}}}}{{{{x}^{{-\frac{1}{2}}}}~}}=\frac{{{{x}^{{\frac{1}{2}}}}~}}{{{{{16}}^{{\frac{1}{2}}}}~{{y}^{{\frac{1}{3}}}}~}}$$

Now, we use the fractional exponent rule and simplify:

$latex =\frac{{\sqrt{x}}}{{\sqrt{{16}}~\sqrt[3]{y}}}$

$latex =\frac{{\sqrt{x}}}{{4~\sqrt[3]{y}}}$