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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