Question

$\frac{ \sqrt{{x}^{3}} \times \sqrt[3]{ {x}^{5} } }{\sqrt[5]{ {x}^{3} }} \times \sqrt[30]{ {x}^{77} }$
please solve giving step-by-step explanation.
Thank You​

Answer 1

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

$\rm :\longmapsto\:\dfrac{ \sqrt{{x}^{3}} \times \sqrt[3]{ {x}^{5} } }{\sqrt[5]{ {x}^{3} }} \times \sqrt[30]{ {x}^{77} }$

We know,

$\boxed{\rm :\longmapsto\: \sqrt[n]{x} = {\bigg( x\bigg) }^{\dfrac{1}{n} } }$

So,

$\rm \:  =  \:  \: \dfrac{{\bigg(x \bigg) }^{\dfrac{3}{2} } \times {\bigg(x \bigg) }^{\dfrac{5}{3} }}{{\bigg(x \bigg) }^{\dfrac{3}{5} }} \times {\bigg(x \bigg) }^{\dfrac{77}{30} }$

We know,

$\boxed{\rm :\longmapsto\: {x}^{m} \times {x}^{n} = {x}^{m + n}}$

$\rm \:  =  \:  \: \dfrac{{\bigg(x \bigg) }^{\dfrac{3}{2} + \dfrac{5}{3}}}{{\bigg(x \bigg) }^{\dfrac{3}{5} }} \times {\bigg(x \bigg) }^{\dfrac{77}{30} }$

$\rm \:  =  \:  \: \dfrac{{\bigg(x \bigg) }^{\dfrac{9 + 10}{6}}}{{\bigg(x \bigg) }^{\dfrac{3}{5} }} \times {\bigg(x \bigg) }^{\dfrac{77}{30} }$

$\rm \:  =  \:  \: \dfrac{{\bigg(x \bigg) }^{\dfrac{19}{6}}}{{\bigg(x \bigg) }^{\dfrac{3}{5} }} \times {\bigg(x \bigg) }^{\dfrac{77}{30} }$

$\boxed{\rm :\longmapsto\: {x}^{m} \div {x}^{n} = {x}^{m - n}}$

$\rm \:  =  \:  \: {\bigg(x \bigg) }^{\dfrac{19}{6} \: - \: \dfrac{3}{5}} \times {\bigg(x \bigg) }^{\dfrac{77}{30} }$

$\rm \:  =  \:  \: {\bigg(x \bigg) }^{\dfrac{95 - 18}{30} \:} \times {\bigg(x \bigg) }^{\dfrac{77}{30} }$

$\rm \:  =  \:  \: {\bigg(x \bigg) }^{\dfrac{77}{30} \:} \times {\bigg(x \bigg) }^{\dfrac{77}{30} }$

$\rm \:  =  \:  \: {\bigg(x \bigg) }^{\dfrac{77}{30} \: + \: \dfrac{77}{30} }$

$\rm \:  =  \:  \: {\bigg(x \bigg) }^{\dfrac{154}{30} \: }$

$\rm \:  =  \:  \: {\bigg(x \bigg) }^{\dfrac{77}{15} \: }$