Question

$\tt{Simplify \: \frac{1}{2} \sqrt{486} - \sqrt{ \frac{27}{ \: \: 2 \: .} } }$
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Answer 1

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$\huge\boxed{\bf{\red{\fcolorbox{blue}{white}{ Answer}}}}$

$\bf \scriptsize{As \: the \: given \: Irrational \: Number \: are \: not \: Similar \: , So \: we \: reduce \: each} \\ \bf \scriptsize { \: irrational \: number \: in \: the \: simplest \: form.}$

$\bf \scriptsize{ \frac{1}{2} \sqrt{486} = \frac{1}{2} \sqrt{9^{2} \times 6 } = \frac{1}{2} \times \sqrt{9^{2} } \times \sqrt{6} }$

$\bf \scriptsize ={ = \frac{1}{2} \times 9 \times \sqrt{6} = \frac{9}{2} \sqrt{6} }$

$\bf \scriptsize{and \: \: \: \: \: \: \: \: \: \: \: \: \: \sqrt{ \frac{27}{2} } = \sqrt{ \frac{54}{4} } = \sqrt{ \frac{3^{2} \times \sqrt{6} }{ \sqrt{2}^{2} } } = \frac{3}{2} \sqrt{6} }$

$\bf \scriptsize{Here, \: \: \frac{1}{2} \sqrt{486} - \sqrt{ \frac{27}{2} } = \frac{9}{2} \sqrt{6} - \frac{3}{2} \sqrt{6} } \\ \\ \bf \scriptsize{ = \sqrt{6} ( \frac{9}{2} - \frac{3}{2}) } \\ \\ \bf \scriptsize \bold{= 3 \sqrt{6} }$

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Answer 2

Given:

Simplify : $\bf{\mathsf{\frac{1}{2} \sqrt{486} - \sqrt{ \frac{27}{2} } }}$

Solution:

$\sf{\therefore \frac{1}{2} \sqrt{486} - \sqrt{\frac{27}{2}}}$

$\implies \sf {\frac{1}{2} \sqrt{2 \times 3 \times 3 \times 3 \times 3 \times 3} - \sqrt{ \frac{3 \times 3 \times 3}{ \sqrt{2} \times \sqrt{2}}}}$

$\implies \sf{\frac{1}{2} \times 3 \times 3 \sqrt{2 \times 3} - \frac{3}{ \sqrt{2} } \sqrt{3} }$

$\implies \sf{\frac{9}{2} \sqrt{6} - \frac{3}{ \sqrt{2} } \sqrt{3}}$

$\implies \sf{\frac{9 \sqrt{6} }{2} - \frac{3 \sqrt{3} }{ \sqrt{2} }}$

$\implies \sf{(\frac{9 \sqrt{6} }{2} \times \frac{ \sqrt{2} }{ \sqrt{2} } ) - ( \frac{3 \sqrt{3} }{ \sqrt{2} } \times \frac{2}{2} )}$

$\implies \sf{\frac{9 \sqrt{12} }{2 \sqrt{2} } - \frac{6 \sqrt{3} }{2 \sqrt{2} } }$

$\implies \sf{ \frac{9 \sqrt{12} - 6 \sqrt{3} }{2 \sqrt{2} } }$

$\bf \sf \implies 3\sqrt{6}$