Question

Find the value of :-
$\boldsymbol{ \longrightarrow\bigg\{ \sqrt[3]{{x}^{4}y} × \dfrac{1}{\sqrt[4]{{x}^{2}{y}^{8}}} \bigg\}^{-6} \times \dfrac{x}{y} }$

$\red \odot \: {x}^{-11} . {y}^{-14}$

$\green \odot \: {x}^{ - 10} . {y}^{ - 13}$

$\pink \odot \: {x}^{ - 4} . {y}^{9}$

$\blue \odot \: {x}^{ - 12} . {y}^{ - 15}$​

Answer 1

Heya!! mate

Your answer is in the attachment

Option third is correct - x^-4 . y^9

Answer 2

Find the value of :-

$\boldsymbol{ \longrightarrow\bigg\{ \sqrt[3]{{x}^{4}y} \times \dfrac{1}{\sqrt[4]{{x}^{2}{y}^{8}}} \bigg\}^{-6} \times \dfrac{x}{y} }$

Identities Used :-

Laws of exponents

$\begin{gathered}(1)\:{\underline{\boxed{\bf{\blue{a^m\times{a^n}\:=\:a^{m\:+\:n}\:}}}}} \\ \end{gathered}$

$\begin{gathered}(2)\:{\underline{\boxed{\bf{\purple{\dfrac{a^m}{a^n}\:=\:a^{m\:-\:n}\:}}}}} \\ \end{gathered}$

$\begin{gathered}(3)\:{\underline{\boxed{\bf{\orange{\dfrac{1}{x^n}\:=\:x^{-n}\:}}}}} \\ \end{gathered}$

$\begin{gathered}(4)\:{\underline{\boxed{\bf{\color{peru}{(a^m)^n\:=\:a^{m\times{n}}\:}}}}} \\ \end{gathered}$

$\begin{gathered}(5)\:{\underline{\boxed{\bf{\red{ {x}^{0} = 1}}}}} \\ \end{gathered}$

$\begin{gathered}(6)\:{\underline{\boxed{\bf{\red{ \sqrt[n]{x} = {\bigg(x\bigg) }^{\dfrac{1}{n} } }}}}} \\ \end{gathered}$

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

Consider

$\boldsymbol{ \sf \bigg\{ \sqrt[3]{{x}^{4}y} \times \dfrac{1}{\sqrt[4]{{x}^{2}{y}^{8}}} \bigg\}^{-6} \times \dfrac{x}{y} }$

$\boldsymbol{ \sf = \bigg\{ {\bigg( {x}^{4}y \bigg) }^{\dfrac{1}{3} } \times \dfrac{1}{ {\bigg( {x}^{2} {y}^{8} \bigg) }^{\dfrac{1}{4}}}\bigg\}^{-6} \times \dfrac{x}{y} }$

$\boldsymbol{ \sf = \bigg\{{ {x}^{ \frac{4}{3}} \times {y}^{ \frac{1}{3}} \times \dfrac{1}{ {x}^{ \frac{1}{2}} \times {y}^{2}}}\bigg\}^{-6} \times \dfrac{x}{y} }$

$\boldsymbol{ \sf = \bigg\{{ {x}^{ \frac{4}{3} - \frac{1}{2} } \times {y}^{ \frac{1}{3} - 2}}\bigg\}^{-6} \times \dfrac{x}{y} }$

$\boldsymbol{ \sf = \bigg\{{ {x}^{ \frac{8 - 3}{6}} \times {y}^{ \frac{1 - 6}{3}}}\bigg\}^{-6} \times \dfrac{x}{y} }$

$\boldsymbol{ \sf = \bigg\{{ {x}^{ \frac{5}{6}} \times {y}^{ \frac{- 5}{3}}}\bigg\}^{-6} \times \dfrac{x}{y} }$

$\boldsymbol{ \sf = \bigg\{{ {x}^{ \frac{5}{6} \times ( - 6)} \times {y}^{ \frac{- 5}{3} \times ( - 6)}}\bigg\} \times \dfrac{x}{y} }$

$\boldsymbol{ \sf = \bigg\{{ {x}^{ - 5} \times {y}^{10}}\bigg\} \times \dfrac{x}{y} }$

$\boldsymbol{ \sf = \bigg\{{ {x}^{ - 5 + 1} \times {y}^{10 - 1}}\bigg\}}$

$\boldsymbol{ \sf = \bigg\{{ {x}^{ - 4} \times {y}^{9}}\bigg\}}$

Hence,

• Option (c) is correct.