# Rational exponent

In this post we will look at the concept of rational exponent with solved examples.

## Rational exponent definition

The number in which power is in the form of rational number is called rational exponent.

Consider the number \mathtt{( 5)^{^{\frac{3}{4}}}} .

Note that the exponent is a rational number 3/4.

The numerator of rational number “3” represents the power.

The denominator of rational number “4” represents the root of the base.

So, the above exponent function can be expressed as;

\mathtt{( 5)^{^{\frac{3}{4}}} =\ \sqrt[4]{5^{3}}}

Hence, the Rational exponent can be generally expressed as;

\mathtt{\ ( a)^{^{\frac{b}{c}}} =\ \sqrt[c]{a^{b}}}

Note:
Here we first put the power and then the number is placed inside the root.

## How to solve rational exponent ?

To solve the rational exponent, follow the below steps;

(a) First express the number in form of power and root.

\mathtt{\ ( a)^{^{\frac{b}{c}}} =\ \sqrt[c]{a^{b}}}

(b) find the value of power by multiplying number by itself.

(c) now take the root of the given number.

I hope you understood the process. Let us see some solved examples for further clarity.

Example 01
Solve \mathtt{( 8)^{^{\frac{2}{3}}}}

Solution

Expressing the number in form of power and root.

\mathtt{( 8)^{^{\frac{2}{3}}} =\ \sqrt[3]{8^{2}}}

Now find the square of number 2.

\mathtt{\Longrightarrow \ \sqrt[3]{8^{2}}}\\\ \\ \mathtt{\Longrightarrow \sqrt[3]{64} \ }

Now find cube root of number 64.

\mathtt{\Longrightarrow \sqrt[3]{64} \ }\\\ \\ \mathtt{\Longrightarrow \ \sqrt[3]{4^{3}}}\\\ \\ \mathtt{\Longrightarrow \ 4}

Hence, 4 is the value of given rational exponent.

Example 02
Solve \mathtt{( 9)^{\frac{2}{4}}}

Solution

Express the rational exponent in form of power & root.

\mathtt{\Longrightarrow ( 9)^{\frac{2}{4}}}\\\ \\ \mathtt{\Longrightarrow \ \sqrt[4]{9^{2}}}

Now find the value of square.

\mathtt{\Longrightarrow \ \sqrt[4]{9^{2}}}\\\ \\ \mathtt{\Longrightarrow \sqrt[4]{9\times 9} \ }\\\ \\ \mathtt{\Longrightarrow \ \sqrt[4]{81}}

Now find the value of root.

\mathtt{\Longrightarrow \ \sqrt[4]{81}}\\\ \\ \mathtt{\Longrightarrow \ \sqrt[4]{3^{4}}}\\\ \\ \mathtt{\Longrightarrow \ 3}

Hence, 3 is the value of given expression.

Example 03
Find value of \mathtt{( 32)^{\frac{1}{5}}}

Solution
Here the power is 1 and root is 5.

Solving the expression.

\mathtt{\Longrightarrow ( 32)^{\frac{1}{5}}}\\\ \\ \mathtt{\Longrightarrow \ \sqrt[5]{32^{1}}}\\\ \\ \mathtt{\Longrightarrow \sqrt[5]{32} \ }\\\ \\ \mathtt{\Longrightarrow \ \sqrt[5]{2^{5}}}\\\ \\ \mathtt{\Longrightarrow \ 2}

Hence, 2 is the solution of given expression.

Example 04
Solve \mathtt{( 125)^{\frac{4}{3}}}

Solution
We will try to solve this questions with different technique.

We know that;

\mathtt{125\ =\ 5\times 5\times 5}\\\ \\ \mathtt{125\ =\ 5^{3}}

So the above expression can be written as;

\mathtt{\Longrightarrow ( 125)^{\frac{4}{3}}}\\\ \\ \mathtt{\Longrightarrow \ \left( 5^{3}\right)^{\frac{4}{3}}}

Now using the power of power rule of exponent.

\mathtt{\Longrightarrow \ \left( 5^{3}\right)^{\frac{4}{3}}}\\\ \\ \mathtt{\Longrightarrow 5^{3\ \times \frac{4}{3}}}\\\ \\ \mathtt{\Longrightarrow \ 5^{\cancel{3} \times \frac{4}{\cancel{3}}}}\\\ \\ \mathtt{\Longrightarrow \ 5^{4}}\\\ \\ \mathtt{\Longrightarrow \ 625}

Hence, 625 is the solution.

Example 05
Find the value of \mathtt{\left(\frac{32}{3125}\right)^{\frac{2}{5}}}

Solution
The above expression can be written as;

\mathtt{\Longrightarrow \left(\frac{32}{3125}\right)^{\frac{2}{5}}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{2^{5}}{5^{5}}\right)^{\frac{2}{5}}}\\\ \\ \mathtt{\Longrightarrow \ \left(\left(\frac{2}{5}\right)^{5}\right)^{\frac{2}{5}}}

Now using power of power rule.

\mathtt{\Longrightarrow \ \left(\left(\frac{2}{5}\right)^{5}\right)^{\frac{2}{5}}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{2}{5}\right)^{5\times \frac{2}{5}}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{2}{5}\right)^{\cancel{5} \times \frac{2}{\cancel{5}}}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{2}{5}\right)^{2}}\\\ \\ \mathtt{\Longrightarrow \ \frac{4}{25}}

Hence, 4/25 is the solution.

Example 06
Find the value of \mathtt{\left(\frac{64}{144}\right)^{\frac{-3}{2}}}

Solution
First convert the negative number into positive by taking exponent of the base.

\mathtt{\Longrightarrow \ \left(\frac{64}{144}\right)^{\frac{-3}{2}}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{144}{64}\right)^{\frac{3}{2}}}

Now we have positive exponent, try to simplify the base.

\mathtt{\Longrightarrow \ \left(\frac{144}{64}\right)^{\frac{3}{2}}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{12^{2}}{8^{2}}\right)^{\frac{3}{2}}}\\\ \\ \mathtt{\Longrightarrow \ \left(\left(\frac{12}{8}\right)^{2}\right)^{\frac{3}{2}}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{12}{8}\right)^{2\times \frac{3}{2}}}\\\ \\ \mathtt{\Longrightarrow \ \left(\frac{12}{8}\right)^{3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{1728}{512}}

Hence, 1728 / 512 is the solution.

Example 07
Find the value of \mathtt{\ \sqrt{1331^{\frac{2}{3}}}}

Solution
The square root is basically a power of 1/2.

The above expression can be written as;

\mathtt{\Longrightarrow \ \sqrt{1331^{\frac{2}{3}}}}\\\ \\ \mathtt{\Longrightarrow \ 1331^{\frac{2}{3} \times \frac{1}{2}}}\\\ \\ \mathtt{\Longrightarrow ( 1331)^{\frac{\cancel{2}}{3} \times \frac{1}{\cancel{2}}}}\\\ \\ \mathtt{\Longrightarrow \ ( 1331)^{\frac{1}{3}}}\\\ \\ \mathtt{\Longrightarrow \ \sqrt[3]{1331}}\\\ \\ \mathtt{\Longrightarrow \ 11}

Hence, 11 is the solution of given expression.

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