Question

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Answer 1

$\large\underline{\bold{Given \:Question - }}$

$\sf \: Simplify : \: {\bigg(\dfrac{256}{625} \bigg) }^{\dfrac{ - 3}{4} } \times {\bigg(\dfrac{81}{16} \bigg) }^{\dfrac{ - 3}{2} } \div {\bigg(\dfrac{9}{2} \bigg) }^{ - 3}$

Answer

Basic Identities Used :-

$1. \: \: \: \: \boxed{ \red{\bf \: {a}^{ - n} = \dfrac{1}{ {a}^{n} }}}$

$2. \: \: \: \: \boxed{ \red{\bf \: {\bigg( { {(a}^{n} )}^{m}\bigg) } = {a}^{mn} }}$

$3. \: \: \: \: \boxed{ \red{\bf \:{\bigg(\dfrac{x}{y} \bigg) }^{m} = \dfrac{ {x}^{m} }{ {y}^{m} } }}$

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

$\rm :\longmapsto\:\sf \:\: {\bigg(\dfrac{256}{625} \bigg) }^{\dfrac{ - 3}{4} } \times {\bigg(\dfrac{81}{16} \bigg) }^{\dfrac{ - 3}{2} } \div {\bigg(\dfrac{9}{2} \bigg) }^{ - 3}$

$\bigg \{\rm \: 256 = 4 \times 4 \times 4 \times 4 = {4}^{4} \\ \rm \: 625 = 5 \times 5 \times 5 \times 5 = {5}^{4} \\ 81 = 3 \times 3 \times 3 \times 3 = {3}^{4} \\ 16 = 2 \times 2 \times 2 \times 2 = {2}^{4} \bigg \}$

$\sf \: \: = \: \: {\bigg(\dfrac{ {4}^{4} }{ {5}^{4} } \bigg) }^{\dfrac{ - 3}{4} } \times {\bigg(\dfrac{ {3}^{4} }{ {2}^{4} } \bigg) }^{\dfrac{ - 3}{2} } \div {\bigg(\dfrac{2}{9} \bigg) }^{3}$

$\sf \: \: = \: \: {\bigg(\dfrac{4}{5} \bigg) }^{4 \times \dfrac{ - 3}{4} } \times {\bigg(\dfrac{3}{2} \bigg) }^{4 \times \dfrac{ - 3}{2} } \div {\bigg(\dfrac{2}{9} \bigg) }^{3}$

$\sf \: \: = \: \: {\bigg(\dfrac{4}{5} \bigg) }^{ - 3} \times {\bigg(\dfrac{3}{2} \bigg) }^{ - 6} \div {\bigg(\dfrac{2}{9} \bigg) }^{3}$

$\sf \: \: = \: \: {\bigg(\dfrac{5}{4} \bigg) }^{3} \times {\bigg(\dfrac{2}{3} \bigg) }^{6} \times {\bigg(\dfrac{9}{2} \bigg) }^{3}$

$\sf \: \: = \: \: {\bigg(\dfrac{5}{ {2}^{2} } \bigg) }^{3} \times {\bigg(\dfrac{2}{3} \bigg) }^{6} \times {\bigg(\dfrac{ {3}^{2} }{2} \bigg) }^{3}$

$\sf \: \: = \: \: {\bigg(\dfrac{ {5}^{3} }{ {2}^{6} } \bigg) } \times {\bigg(\dfrac{ {2}^{6} }{ {3}^{6} } \bigg) } \times {\bigg(\dfrac{ {3}^{6} }{ {2}^{3} } \bigg) }$

$\sf \: \: = \: \: {\bigg(\dfrac{ {5}^{3} }{ \cancel{{2}^{6}} } \bigg) } \times {\bigg(\dfrac{ \cancel{ {2}^{6}} }{ \cancel{ {3}^{6}} } \bigg) } \times {\bigg(\dfrac{ \cancel{{3}^{6}} }{ {2}^{3} } \bigg) }$

$\sf \: \: = \: \: \dfrac{ {5}^{3} }{ {2}^{3} }$

$\sf \: \: = \: \: \dfrac{5 \times 5 \times 5}{2 \times 2 \times 2}$

$\sf \: \: = \: \: \dfrac{125}{8}$

Additional Information :-

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

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

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

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

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

Answer 2

$\huge\underline{\underline {\sf{ Question }}} :$

• Simplify ${\bigg( \dfrac{256}{625}\bigg)}^{ \frac{ - 3}{ \: \: 4 } } \times {\bigg( \dfrac{81}{16}\bigg)}^{ \frac{ - 3}{ \: \: 2} } \div {\bigg(\dfrac{9}{2}\bigg)}^{ - 3}$

$\huge\underline{\underline {\sf{ Solution }}} :$

$\sf \dashrightarrow \ \frac{{ \bigg(\dfrac{ {4}^{4} }{ {5}^{4} } \bigg)}^{ \frac{ - 3}{ \: 4} } \times {\bigg( \dfrac{ {3}^{4} }{ {2}^{4} } \bigg)}^{ \frac{ - 3}{ \: \: 2} } }{{ \bigg(\dfrac{2}{9}\bigg)}^{3} }$

$\sf \dashrightarrow \ \frac{{ \bigg(\dfrac{ 4 }{ 5 } \bigg)}^{ - 3 } \times {\bigg( \dfrac{ 3 }{ 2 } \bigg)}^{ - 6 } }{{ \bigg(\dfrac{2}{9}\bigg)}^{3} }$

$\sf \dashrightarrow \ \frac{{ \bigg(\dfrac{ 5 }{ 4 } \bigg)}^{ 3 } \times {\bigg( \dfrac{ 2 }{ 3 } \bigg)}^{ 6 } }{{ \bigg(\dfrac{2}{9}\bigg)}^{3} }$

$\sf \dashrightarrow \ \dfrac{ { \: 5 \: }^{3} }{ \cancel{ { 2 }^{6} } } \times \dfrac{ \cancel{ {2}^{6} } }{\cancel{ {3}^{6} } } \times \dfrac{ \cancel{{3}^{6}} }{ {2}^{3} }$

$\sf \dashrightarrow \ \dfrac{ { \: 5 \: }^{3} }{ { 2 }^{3} }$

$\sf \dashrightarrow \ \dfrac{ 125 }{ 8 }$

Hence ,

• The answer is $\sf \ \dfrac{ 125 }{ 8 }$

$\huge\underline{\underline {\sf{ Extra \: Information }}} :$

$\large\dashrightarrow\boxed{\sf{ {a}^{m} \times {a}^{n} = {a}^{m + n} }}$

$\large\dashrightarrow\boxed{\sf{ \frac{{a}^{m}}{{a}^{n}} = {a}^{m - n} }}$

$\large\dashrightarrow\boxed{\sf{ {a}^{0} = 1 }}$

$\large\dashrightarrow\boxed{\sf{ { a }^{ -n } = \dfrac { 1 } { { a }^{ n } } }}$