Question

Find the square roots of 3364 and 1936 and hence find the value of
$\frac{ \sqrt{0.3364} + \sqrt{0.1936} }{ \sqrt{0.3364} - \sqrt{0.1936} }$
​

Answer 1

To find :

• Square roots of 3364 & 1936.

• The value of $\sf { \dfrac{ \sqrt{0.3364} + \sqrt{0.1936} }{ \sqrt{0.3364} - \sqrt{0.1936} }}$

Calculation :

Let us calculate the square root of 3364 and 1936 by prime factorization.

$\begin {array}{c | c} \sf{2}& \underline{ \sf{3364}} \\\sf{2}& \underline{ \sf{1682}} \\ \sf{29}& \underline{ \sf{ \: 841 \: }} \\ \sf{29}& \underline{ \sf{ \: \: 29 \: \: }} \\ \: & \: \sf{1}\end{array}$

By prime factorization, we get :-

$\sf{\longrightarrow \: 3364 = 2 \times 2 \times 29 \times 29 } \\ \\ \sf{\longrightarrow \: 3364 = {2}^{2} \times {29}^{2} } \\ \\ \sf{\longrightarrow \: \sqrt{3364} = \sqrt{ {2}^{2} \times {29}^{2} } } \\ \\ \sf{\longrightarrow \: \sqrt{3364} =2 \times 29} \\ \\ \sf \red{\longrightarrow \: \sqrt{3364} =58}$

Henceforth, square root of √3364 is 58.

Also,

$\begin {array}{c | c} \sf{2}& \underline{ \sf{1936}} \\\sf{2}& \underline{ \sf{968}} \\ \sf{2}& \underline{ \sf{ \: 484 \: }} \\ \sf{2}& \underline{ \sf{ \: \: 242 \: \: }} \\\sf{2}& \underline{ \sf{121}} \\ \sf{11} & \underline{ \sf{11}} \\ \: & \: \sf{1}\end{array}$

$\sf{\longrightarrow \: 1936 = 2 \times 2 \times 2 \times 2 \times 11 \times 11 } \\ \\ \sf{\longrightarrow \: 1936 = {2}^{2} \times {2}^{2} \times {11}^{2} } \\ \\ \sf{\longrightarrow \: \sqrt{1936} = \sqrt{ {2}^{2} \times {2}^{2} \times {11}^{2} } } \\ \\ \sf{\longrightarrow \: \sqrt{1936} =2 \times 2 \times 11} \\ \\ \sf \red{\longrightarrow \: \sqrt{1936} =44}$

Henceforth, square root of √1936 is 44.

Now,

• The value of $\sf { \dfrac{ \sqrt{0.3364} + \sqrt{0.1936} }{ \sqrt{0.3364} - \sqrt{0.1936} }}$

Let us tackle this by dividing into two parts. First one is numerator and second one is denominator.

$\sf { \longrightarrow \: \dfrac{ \sqrt{0.3364} + \sqrt{0.1936} }{ \sqrt{0.3364} - \sqrt{0.1936} }} \\ \\ \\ \sf { \longrightarrow \: \dfrac{ \sqrt{ \cfrac{3364}{10000} } + \sqrt{ \cfrac{1936}{10000} } }{ \sqrt{ \cfrac{3364}{10000} } - \sqrt{ \cfrac{1936}{10000} } }}$

Solving numerator first.

$\sf { \longrightarrow \: \sqrt{ \cfrac{3364}{10000} } + \sqrt{ \cfrac{1936}{10000} } } \\ \\ \sf { \longrightarrow \: \dfrac{ \sqrt{3364} }{ \sqrt{10000} } + \dfrac{ \sqrt{1936} }{ \sqrt{10000} } }$

We got the square root of 3364 and 1936 in first part of the question.

Square root of 10000 is 100 as the number of zeroes at the end of the number in a perfect square is always get doubled. [ Like $\sf { {10}^{2} = 100}$.

$\sf { \longrightarrow \: \dfrac{ 58 }{ 100 } + \dfrac{ 44}{ 100 } }$

$\sf { \: \dfrac{ 58 }{ 100 } + \dfrac{ 44}{ 100 } } \\ \\ \sf { \longrightarrow \: 0.58 + 0.44 \: \: } \\ \\ \sf{ \longrightarrow \: 1.02 \: \: \: (Numerator)}$

Solving denominator.

$\sf { \longrightarrow \: \sqrt{ \cfrac{3364}{10000} } - \sqrt{ \cfrac{1936}{10000} } } \\ \\ \sf { \longrightarrow \: \dfrac{ \sqrt{3364} }{ \sqrt{10000} } - \dfrac{ \sqrt{1936} }{ \sqrt{10000} } } \\ \\ \sf { \longrightarrow \: \dfrac{58}{100} - \dfrac{44}{100} } \\ \\ \sf { \longrightarrow \:0.58 + 0.44} \\ \\ \sf { \longrightarrow \:0.14 \: \: \: (denominator)}$

Thus,

$\sf { \longrightarrow \: \dfrac{ \sqrt{0.3364} + \sqrt{0.1936} }{ \sqrt{0.3364} - \sqrt{0.1936} }} \\ \\ \\ \sf { \longrightarrow \: \dfrac{ \sqrt{ \cfrac{3364}{10000} } + \sqrt{ \cfrac{1936}{10000} } }{ \sqrt{ \cfrac{3364}{10000} } - \sqrt{ \cfrac{1936}{10000} } }} \\ \\ \sf { \longrightarrow \: \dfrac{1.02}{0.14} } \\ \\ \sf { \longrightarrow \: \dfrac{10200}{1400} } \\ \\ \sf \red { \longrightarrow \: \dfrac{102}{14} }$

Henceforth, the value of $\sf { \dfrac{ \sqrt{0.3364} + \sqrt{0.1936} }{ \sqrt{0.3364} - \sqrt{0.1936} }}$ is $\sf{\dfrac{120}{14} }$