Question

please give step by step explanation:
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Answer 1

$\underline {\Large {\bf { Answer : }}}$

Find the square root of :

• $\sf { 5 \dfrac{580}{729} }$

$\sf { \longrightarrow \sqrt{ 5 \dfrac{580}{729}} }$

Solution:

Here, firstly we'll convert the mixed fraction into proper fraction. So,

$\sf { \longrightarrow \sqrt{ 5 \dfrac{580}{729}} = \sqrt{ \dfrac{729 \times 5 + 580}{729} } }$

$\sf { \longrightarrow \sqrt{ 5 \dfrac{580}{729}} = \sqrt{ \dfrac{3645+ 580}{729} } }$

$\sf { \longrightarrow \sqrt{ 5 \dfrac{580}{729}} = \sqrt{ \dfrac{4225}{729} } }$

Now, as we know that :

$\underline{ \boxed {\bf \gray { \sqrt{ \dfrac{ x }{y} } = \dfrac{ \sqrt{x} }{ \sqrt{y} } }}}$

$\sf { \longrightarrow \sqrt{ \dfrac{4225}{729} } = \dfrac{ \sqrt{4225} }{ \sqrt{729} } }$

Calculating the value of √4225 :

By prime factorization :

$\begin{array}{c | c} \sf5& \sf \underline{4225} \\ \sf5& \sf \underline{ \: 845 \: } \\ \sf13& \sf \underline{ \: 169 \: } \\ \sf13& \sf \underline{ \: \: 13 \: \: } \\ \sf& \sf 1 \\ \\ \end{array}$

→ 4225 = 5 × 5 × 13 × 13

→ 4225 = 5² × 13²

→ √4225 = √5² × 13²

→ √4225 = 65

Calculating the value of √729 :

By prime factorization :

$\begin{array}{c | c} \sf3& \sf \underline{729} \\ \sf3& \sf \underline{ \: 243 \: } \\ \sf3& \sf \underline{ \: 81\: } \\ \sf3& \sf \underline{ \: \: 27 \: \: } \\ \sf3& \sf \underline{ \: \: 9 \: \: } \\ \sf3& \sf \underline{ \: \: 3 \: \: } \\ \sf& \sf 1 \\ \\ \end{array}$

→ 729 = 3 × 3 × 3 × 3 × 3 × 3

→ 729 = 3² × 3² × 3²

→ √729 = √3² × 3² × 3²

→ √729 = 3 × 3 × 3

→ √729 = 27

So,

$\sf { \longrightarrow \dfrac{ \sqrt{4225} }{ \sqrt{729} } = \dfrac{65}{27} }$

Therefore,

$\boxed{\sf\red { \longrightarrow \sqrt{ \dfrac{4225}{729} } = \dfrac{65}{27} }}$