Question

Find the value of the given question below




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Answer 1

Answer:

if you find helpful please mark as brilliant

Answer 2

Question

• $\\ \bf \bigstar { \: \dfrac{ {2}^{n + 2} \times ({ {2}^{n - 1}) }^{n + 1} }{ {2}^{n(n - 1)} } \div {4}^{n} }$

$\\\text{\bf Step by step explanation}$

${ \sf ~~~~~:\implies~~ \dfrac{ {2}^{n + 2} \times ({ {2}^{n - 1}) }^{n + 1} }{ {2}^{n(n - 1)} } \div {4}^{n} }$

${ \sf ~~~~~:\implies~~ \dfrac{ {2}^{n + 2} \times ( {2}^{(n - 1)(n + 1)}) }{ {2}^{n \times n -n \times 1} } \div ({ {2}^{2} })^{n} }$

• $\small\text{Using identity : \tiny \gray{(a-b)(a+b) = a² \: -b²}}$

${ \sf ~~~~~:\implies~~ \dfrac{ {2}^{n + 2} \times {2}^{ {n}^{2} - {1}^{2} } }{ {2}^{ {n}^{2} - n} } \div {2}^{2n} }$

${ \sf ~~~~~:\implies~~ \dfrac{ {2}^{n + 2} \times {2}^{ {n}^{2} - 1 } }{ {2}^{ {n}^{2} - n} } \div {2}^{2n} }$

${ \sf ~~~~~:\implies~~ \dfrac{ {2}^{n + 2 + {n}^{2} - 1 } }{ {2}^{ {n}^{2} } \times {2}^{ - n} } \div {2}^{2n} }$

${ \sf ~~~~~:\implies~~ \dfrac{ {2}^{ {n}^{2} + n + 1 } }{ {2}^{ {n}^{2} } \times \dfrac{1}{ {2}^{n} } } \div {2}^{2n} }$

${ \sf ~~~~~:\implies~~ \dfrac{ {2}^{ {n}^{2} + n + 1 } }{ \dfrac{ {2}^{ {n}^{2} } }{ {2}^{n} } } \div {2}^{2n} }$

${ \sf ~~~~~:\implies~~ \left({2}^{ {n}^{2} + n + 1 } \times \dfrac{ {2}^{n} }{ {2}^{ {n}^{2} } } \right) \div {2}^{2n} }$

${ \sf ~~~~~:\implies~~ \left({2}^{ \cancel{ {n}^{2}} + n + 1 + n \: \cancel{ - {n}^{2}} } \right) \div {2}^{2n} }$

${ \sf ~~~~~:\implies~~ {2}^{ 2n + 1 } \div {2}^{2n} }$

${ \sf ~~~~~:\implies~~ {2}^{ 1 + \cancel{2n} \cancel{ - 2n } } }$

${ \sf ~~~~~:\implies~~ 2 }$

$\sf ~~~~~♥~~~~~§ { \: \dfrac{ {2}^{n + 2} \times ({ {2}^{n - 1}) }^{n + 1} }{ {2}^{n(n - 1)} } \div {4}^{n} = \orange{\bf{2} }}$

More to know:-

$\red{ \tiny\begin{gathered}\begin{gathered}\\\\\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered} \bigstar \: \underline{\bf{}}\\ {\boxed{\begin{array}{c | c} \frac{ \: ~~~~~~~~~~\: \: \: \: \:\sf Laws \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: }{ } &\frac{ \: ~~~~~~~~~~ \: \: \: \: \:\sf Example \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: }{}\\ \sf \bigstar{a}^{m} \times {a}^{n} = {a}^{m + n} & \sf {a}^{2} \times {a}^{3} = {a}^{2 + 3} = {a}^{6} \\ \\ \sf \bigstar{a}^{m} \div {a}^{n} = {a}^{m - n}& \sf {a}^{3} \div {a}^{2} = {a}^{3 - 2} = {a}^{1} \\ \\ \sf{\bigstar \: \: \: \: \: \: ( {a}^{m} ) ^{n} = {a}^{mn} } & \sf( {a}^{2} ) ^{3} = {a}^{2 \times 3} = {a}^{6} \\ \\ {\bigstar\sf a {}^{m} \times {n}^{m} = (ab) ^{m} } &\sf a {}^{2} \times {b}^{2} = (ab) ^{2}\\ \\ \sf\bigstar \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: {a}^{0} = 1& \sf {2}^{0} = 1 \: \: \: \: \\ \\ \sf \bigstar \: \: \: \: {\dfrac{ {a}^{m} }{ {b}^{m} }= \left( \dfrac{a}{b} \right) ^{m} }& \sf{\dfrac{ {a}^{2} }{ {b}^{2} }= \left( \dfrac{a}{b} \right) ^{2} }\\\\\bigstar~~~~~~~ \sf x^{\frac{m}{n} }=\sqrt[n]{x^m}\sf = (\sqrt[n]{x})^m & \sf x^{\frac{2}{3} }=\sqrt[3]{x^2} = (\sqrt[n]{x})^m\\ \\\\ \end{array}}}\end{gathered}\end{gathered}\end{gathered}\end{gathered}\end{gathered}\end{gathered}}$