Question

Find the value of :-


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

​

Answer 1

Answer:

Answer = x^-4/ y^-9

The simplification is done correctly.

Answer 2

Given :

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

Identities Used :

• Laws of exponents

• ${\underline{\boxed{\sf{\blue{a^m\times{a^n}\:=\:a^{m\:+\:n}\:}}}}}$★

• ${\underline{\boxed{\sf{\purple{\dfrac{a^m}{a^n}\:=\:a^{m\:-\:n}\:}}}}}$★

• ${\underline{\boxed{\sf{\orange{\dfrac{1}{x^n}\:=\:x^{-n}\:}}}}}$★

• ${\underline{\boxed{\sf{\color{peru}{(a^m)^n\:=\:a^{m\times{n}}\:}}}}}$★

• ${\underline{\boxed{\bf{\red{ {x}^{0} = 1}}}}}$★

• ${\underline{\boxed{\sf{\green{ \sqrt[n]{x} = {\bigg(x\bigg) }^{\dfrac{1}{n} } }}}}}$★

Solution :

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

${\sf{ :\implies\bigg\{\bigg( {x}^{4}y \bigg)^{\frac{1}{3} } × \dfrac{1}{( {x}^{2} {y}^{8 })^{\frac{1}{4}}}\bigg\}^{-6}~ \times ~\dfrac{x}{y}}}$

${\sf{ :\implies\bigg\{ {x} ^{\frac{4}{3}} \times {y}^{\frac{1}{3}} \times \dfrac{1}{ {x}^{ \frac{1}{2} ~\times ~{y}^{2}}}\bigg\}^{-6} ~\times~ \dfrac{x}{y} }}$

${\sf{ :\implies \bigg\{ {x}^{ \frac{4}{3} - \frac{1}{2}}~ \times~ {y}^{ \frac{1}{3} - 2}\bigg\}^{-6} ~\times ~\dfrac{x}{y} }}$

${\sf{ :\implies\bigg\{ {x}^{ \frac{8~ - ~3}{6}} ~\times ~{y}^{ \frac{1~ - ~6}{3}}\bigg\}^{-6}~ \times~ \dfrac{x}{y} }}$

${\sf{ :\implies\bigg\{ {x}^{ \frac{5}{6}} ~\times~ {y}^{ \frac{- 5}{3}}\bigg\}^{-6} ~\times~ \dfrac{x}{y} }}$

${\sf{ :\implies\bigg\{ {x}^{ \frac{5}{6}}~ \times ~( - 6) ~\times~ {y}^{ \frac{- 5}{3}}~ \times~ ( - 6)\bigg\} ~\times~ \dfrac{x}{y} }}$

${\sf{ :\implies\bigg\{ {x}^{ - 5} ~\times~ {y}^{10}\bigg\} ~\times~ \dfrac{x}{y} }}$

${\sf{ :\implies\bigg\{ {x}^{ - 5 ~+~ 1} ~\times~ {y}^{10~ - ~1}}\bigg\}}$

• $\boxed{\sf{\frak{\pink{\bigg\{ {x}^{ - 4}~ \times ~{y}^{9}}\pink{\bigg\}}}}}$★

$~$

Hence,

• The value is ${\sf{\bigg\{{ {x}^{ - 4} \times {y}^{9}}\bigg\}}}$