Question

find the value of ;-
$\sqrt{432 } \times \sqrt{8748}$
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Answer 1

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

Consider

$\green{ \boxed{ \bf \: Prime \: factorization \: of \: 432}}$

$\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:432\: \:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:216\:\:\:}} \\ {\underline{\sf{2}}}& \underline{\sf{\:\:108\:\:\:}} \\ {\underline{\sf{2}}}& \underline{\sf{\:\:54\:\:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:27\:\:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:9\:\:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:3\:\:\:}} \\{\sf{}}&\underline{\sf{\:\:1\ \:\:}}\end{array}\end{gathered}\end{gathered}\end{gathered}$

$\green{ \boxed{ \bf \: Prime \: factorization \: of \: 432 = {2}^{4} \times {3}^{3}}}$

Consider,

$\green{ \boxed{ \bf \: Prime \: factorization \: of \: 8748}}$

$\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:8748\: \:\:}}}\\ {\underline{\sf{2}}}& \underline{\sf{\:\:4374\:\:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:2187\:\:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:729\:\:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:243\:\:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:81\:\:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:27\:\:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:9\:\:\:}} \\ {\underline{\sf{3}}}& \underline{\sf{\:\:3\:\:\:}} \\{\sf{}}&\underline{\sf{\:\:1\ \:\:}}\end{array}\end{gathered}\end{gathered}\end{gathered}$

$\green{ \boxed{ \bf \: Prime \: factorization \: of \: 8748 = {2}^{2} \times {3}^{7}}}$

Now,

$\rm :\longmapsto\: \sqrt{432} \times \sqrt{8748}$

$\rm \: = \: \: \sqrt{432 \times 8748}$

$\rm \: = \: \: \sqrt{ {2}^{4} \times {3}^{3} \times {2}^{2} \times {3}^{7} }$

$\rm \: = \: \: \sqrt{ {2}^{6} \times {3}^{10} }$

$\rm \: = \: \: {2}^{3} \times {3}^{5}$

$\rm \: = \: \: 8 \times 243$

.

$\rm \: = \: \: 1944$

Hence,

$\: \: \: \: \: \: \: \: \: \: \: \underbrace{\boxed{\bf \: \sqrt{432} \times \sqrt{8748} = 1944 \: }}$

Additional Information :-

$\boxed{ \sf \: \sqrt{ {x}^{2} } = x}$

$\boxed{ \sf \: \sqrt{ {x}^{4} } = {x}^{2} }$

$\boxed{ \sf \: \sqrt{x \times y} = \sqrt{x} \times \sqrt{y}}$

$\boxed{ \sf \: \sqrt{\dfrac{x}{y} } = \dfrac{ \sqrt{x} }{ \sqrt{y} }}$

Answer 2

$\begin{gathered}{\Huge{\textsf{\textbf{\underline{\underline{\purple{Answer:}}}}}}}\end{gathered}$

$\implies$Please Drop Some Thanks.