Question

What is square root of 625/144

Answer 1

$\large\underline{\sf{Solution-}}$

$\rm :\longmapsto\: \sqrt{\dfrac{625}{144} }$

So, to find this squareroot, let we first find the prime factorization of 144 and 625.

Consider,

$\red{\bf :\longmapsto\:Prime \: factorization \: of \: 625}$

$\red{\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{5}}}&{\underline{\sf{\:\:625 \:\:}}}\\ {\underline{\sf{5}}}& \underline{\sf{\:\:125 \:\:}} \\\underline{\sf{5}}&\underline{\sf{\:\:25\:\:}} \\ {\underline{\sf{5}}}& \underline{\sf{\:\:5 \:\:}}\\\underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}\end{gathered}}$

Hence,

$\red{\bf :\longmapsto\:Prime \: factorization \: of \: 625 = 5 \times 5 \times 5 \times 5}$

Now, Consider

$\green{\bf :\longmapsto\:Prime \: factorization \: of \: 144}$

$\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:144 \:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:72 \:\:}} \\\underline{\sf{2}}&\underline{\sf{\:\:36\:\:}} \\ {\underline{\sf{2}}}& \underline{\sf{\: \: 18 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:9 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:3 \:\:}}\\\underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}\end{gathered}$

Hence,

$\green{\bf :\longmapsto\:Prime \: factorization \: of \: 144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3}$

Therefore,

$\purple{\rm :\longmapsto\: \sqrt{\dfrac{625}{144} } }$

$\purple{\rm \: = \: \sqrt{\dfrac{ \underbrace{5 \times 5} \times \underbrace{5 \times 5}}{\underbrace{2 \times 2} \times \underbrace{2 \times 2} \times \underbrace{3 \times 3}} } }$

$\purple{\rm \: = \:\dfrac{5 \times 5}{2 \times 2 \times 3} }$

$\purple{\rm \: = \:\dfrac{25}{12} }$

Thus,

$\pink{\bf\implies \:\boxed{ \tt{ \: \sqrt{ \frac{625}{144} } = \frac{25}{12} \: \: }}}$

Answer 2

Answer:

$\large\underline{\sf{Solution-}}$

$\rm :\longmapsto\: \sqrt{\dfrac{625}{144} }$

So, to find this squareroot, let we first find the prime factorization of 144 and 625.

Consider,

$\red{\bf :\longmapsto\:Prime \: factorization \: of \: 625}$

$\red{\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{5}}}&{\underline{\sf{\:\:625 \:\:}}}\\ {\underline{\sf{5}}}& \underline{\sf{\:\:125 \:\:}} \\\underline{\sf{5}}&\underline{\sf{\:\:25\:\:}} \\ {\underline{\sf{5}}}& \underline{\sf{\:\:5 \:\:}}\\\underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}\end{gathered}}$

Hence,

$\red{\bf :\longmapsto\:Prime \: factorization \: of \: 625 = 5 \times 5 \times 5 \times 5}$

Now, Consider

$\green{\bf :\longmapsto\:Prime \: factorization \: of \: 144}$

$\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:144 \:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:72 \:\:}} \\\underline{\sf{2}}&\underline{\sf{\:\:36\:\:}} \\ {\underline{\sf{2}}}& \underline{\sf{\: \: 18 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:9 \:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:3 \:\:}}\\\underline{\sf{}}&{\sf{\:\:1 \:\:}} \end{array}\end{gathered}$

Hence,

$\green{\bf :\longmapsto\:Prime \: factorization \: of \: 144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3}$

Therefore,

$\purple{\rm :\longmapsto\: \sqrt{\dfrac{625}{144} } }$

$\purple{\rm \: = \: \sqrt{\dfrac{ \underbrace{5 \times 5} \times \underbrace{5 \times 5}}{\underbrace{2 \times 2} \times \underbrace{2 \times 2} \times \underbrace{3 \times 3}} } }$

$\purple{\rm \: = \:\dfrac{5 \times 5}{2 \times 2 \times 3} }$

$\purple{\rm \: = \:\dfrac{25}{12} }$

Thus,

$\pink{\bf\implies \:\boxed{ \tt{ \: \sqrt{ \frac{625}{144} } = \frac{25}{12} \: \: }}}$