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**.