Square Root of Number in the Fraction Form

In square root of number in the fraction form, suppose the square root of a fraction $\frac{x}{a}$ is that fraction $\frac{y}{a}$ which when multiplied by itself gives the fraction $\frac{x}{a}$.

If x and y are squares of some numbers then,

$\sqrt{\frac{x}{y}} = \frac{\sqrt{x}}{\sqrt{y}}$

If the fraction is expressed in a mixed form, convert it into improper fraction.

Find the square root of numerator and denominator separately and write the answer in the fraction form.

Examples on square root of number in the fraction form are explained below;

1. Find the square root of $\frac{625}{256}$

Solution:

$\sqrt{\frac{625}{256}} = \frac{\sqrt{625}}{\sqrt{256}}$

Now, we find the square roots of 625 and 256 separately.

Thus, √625 = 25 and √256 = 16

⇒ $\sqrt{\frac{625}{256}} = \frac{\sqrt{625}}{\sqrt{256}}$ = $\frac{25}{26}$

2. Evaluate: $\sqrt{\frac{441}{961}}$.

Solution:

$\sqrt{\frac{441}{961}} = \frac{\sqrt{441}}{\sqrt{961}}$

Now, we find the square roots of 441 and 961 separately.

Thus, √441 = 21 and √961 = 31

⇒ $\sqrt{\frac{441}{961}}$ = $\frac{\sqrt{441}}{\sqrt{961}}$ = $\frac{21}{31}$

3. Find the values of $\sqrt{\frac{7}{2}}$ up to 3 decimal places.

Solution:

To make the denominator a perfect square, multiply the numerator and denominator by √2.

Therefore, $\frac{\sqrt{7} \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}}$ = $\frac{\sqrt{14}}{2}$

Now, we find the square roots of 14 up to 3 places of decimal.

Thus, √14 = 3.741 up to 3 places of decimal.

= 3.74 correct up to 2 places of decimal.

Therefore, $\frac{\sqrt{14}}{2}$ = $\frac{3.74}{2}$ = 1.87.

4. Find the square root of 1$\frac{56}{169}$

Solution:

1$\frac{56}{169}$ = $\frac{225}{169}$

Therefore, $\sqrt{1\frac{56}{169}}$ = $\sqrt{\frac{225}{169}} = \frac{\sqrt{225}}{\sqrt{169}}$

We find the square roots of 225 and 169 separately

Therefore, √225 = 15 and √169 = 13

⇒ $\sqrt{1\frac{56}{169}}$ = $\sqrt{\frac{225}{169}} = \frac{\sqrt{225}}{\sqrt{169}}$ = $\frac{15}{13}$ = 1$\frac{2}{13}$

5. Find the value of $\frac{\sqrt{243}}{\sqrt{363}}$.

Solution:

$\frac{\sqrt{243}}{\sqrt{363}}$ = $\sqrt{\frac{243}{363}}$ = $\sqrt{\frac{81}{121}} = \frac{\sqrt{81}}{\sqrt{121}}$ = $\frac{9}{11}$

6. Find out the value of √45 × √20.

Solution:

√45 × √20 = √(45 × 20)

= √(3 × 3 × 5 × 2 × 2 × 5)

= √(3 × 3 × 2 × 2 × 5 × 5 )

= (3 × 2 × 5)

= 30.