# Square root of fractions

In this chapter we will learn to calculate square root of fraction numbers with solved examples.

We have already covered square root calculation of whole numbers and decimal numbers in previous chapters. Click the red link to understand these concepts.

## How to find square root of fractions ?

Let \mathtt{\frac{a}{b}} is the given fraction.

It’s square root can be expressed as \mathtt{\sqrt{\frac{a}{b}} \ } .

The square root of fraction is equal to individual square root of numerator and denominator.

\mathtt{\sqrt{\frac{a}{b}} \ =\frac{\sqrt{a}}{\sqrt{b}}}

Keeping the above expression in mind, you can calculate the square root using following steps;

(a) Find square root of numerator and denominator separately.

(b) If possible, reduce the solved fraction to its lowest terms.

I hope you understood the above process. Let us solve some problems for further clarity.

Example 01
Find the square root of fraction \mathtt{\ \frac{16}{25}}

Solution
The square root of fraction can be expressed as follows;

\mathtt{\Longrightarrow \sqrt{\frac{16}{25}} \ =\frac{\sqrt{16}}{\sqrt{25}} \ }

Find the square root of numerator and denominator separately.

Here we will use prime factorization method to find square root.

Square root of 16

\mathtt{\sqrt{16} =\sqrt{2^{2} \times 2^{2}}}\\\ \\ \mathtt{\sqrt{16} \ \ =\ 2\ \times \ 2}\\\ \\ \mathtt{\sqrt{16} \ =\ 4}

Square root of 25

\mathtt{\sqrt{25} =\sqrt{5^{2}}}\\\ \\ \mathtt{\sqrt{25} \ \ =\ 5}

Putting the values of square root in both expression.

\mathtt{\sqrt{\frac{16}{25}} \ =\frac{4}{5} \ }

Hence, 4/5 is the final solution.

Example 02
Find the square root of fraction \mathtt{\ \frac{676}{961}}

Solution
The square root can be expressed as;

\mathtt{\Longrightarrow \sqrt{\frac{676}{961}} \ =\frac{\sqrt{676}}{\sqrt{961}}}

Find the square root of numerator and denominator separately.

Here we will use long division method of square root calculation.

Square root of 676

Hence, 26 is the square root of 676.

Calculating square root of 961.

Hence, 31 is the square root of 961.

Putting the values of square root in main expression.

\mathtt{\Longrightarrow \sqrt{\frac{676}{961}} \ =\frac{26}{31} \ }

Hence, 26/31 is the solution.

Example 03
Find square root of fraction \mathtt{\frac{2025}{8649}}

Solution
The square root can be expressed as;

\mathtt{\Longrightarrow \sqrt{\frac{2025}{8649}} \ =\frac{\sqrt{2025}}{\sqrt{8649}} \ }

Calculate square root of numerator 2025

Hence 45 is the square root of 2025.

Calculating square root of denominator 8649.

Hence, 93 is the square root of 8649.

Putting the value of square roots in main expression.

\mathtt{\Longrightarrow \frac{\sqrt{2025}}{\sqrt{8649}} =\ \frac{45}{93} \ }

Hence, 45/93 is the solution.

Example 04
Find the square root of \mathtt{\frac{10}{3}}

Solution
The square root can be expressed as;

\mathtt{\Longrightarrow \ \sqrt{\frac{10}{3}} =\ \frac{\sqrt{10}}{\sqrt{3}}}

Multiply numerator and denominator by \mathtt{\sqrt{3}} so that calculation becomes easier.

\mathtt{\Longrightarrow \ \frac{\sqrt{10} \ \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}}}\\\ \\ \mathtt{\Longrightarrow \ \frac{\sqrt{30}}{3}}

Calculate the value of \mathtt{\sqrt{30}} using long division method.

Hence, 5.47 is the approximate square root of number 30.

Putting the value in main square root expression.

\mathtt{\Longrightarrow \ \frac{5.47}{3} \ =\ 1.823}

Hence, 1.823 is the solution of given expression.

Example 05
Find the square root of \mathtt{\frac{10404}{7396}}

Solution
The square root can be expressed as;

\mathtt{\Longrightarrow \sqrt{\frac{10404}{7396}} =\frac{\sqrt{10404}}{\sqrt{7396}}}

Find the value of square root of numerator and denominator separately.

Calculating square root of 10404.

Hence, 102 is the square root of 10404

Now find the square root of 7396

Hence, 86 is the square root of 7396.

Putting the values of square root in main expression.

\mathtt{\Longrightarrow \sqrt{\frac{10404}{7396}} =\frac{102}{86} \ }

Hence, 102/86 is the solution of given fraction.

The solution can be further reduced by dividing numerator and denominator by 2.

\mathtt{\Longrightarrow \ \frac{102}{86}}\\\ \\ \mathtt{\Longrightarrow \ \frac{102\div 2}{86\div 2}}\\\ \\ \mathtt{\Longrightarrow \ \frac{51}{43}}

Hence, 51/43 is the final solution.