Question

Solve this question with explanation ⬆⬆
$\huge{ \frac{ {2}^{ - n}\times {8}^{2n + 1} \times {16}^{2n} } { {4}^{3n} } }$

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Answer 1

Question:

Solve :

$\sf\dfrac{2^{-n}\times8^{2n+1}\times16^{2n}}{4^{3n}}$

Solution:

$\sf\dfrac{2^{-n}\times8^{2n+1}\times16^{2n}}{4^{3n}}$

Use Identity

$\sf\green{(a^m)^n=a^{mn}}$

$\sf\implies\dfrac{2^{-n}\times[(2)^3]^{2n+1}\times[(2^4)]^{2n}}{[(2^2)]^{3n}}$

$\sf\implies\dfrac{2^{-n}\times2^{6n+3}\times2^{8n}}{2^{6n}}$

Use Identity

$\sf\purple{a^m\times\:a^n=a^{m+n}}$

$\sf\implies\dfrac{2^{-n+6n+3+8n}}{2^{6n}}$

$\sf\implies\dfrac{2^{13n+3}}{2^{6n}}$

Use Identity

$\blue{\dfrac{a^m}{a^n}=a^{m-n}}$

$\sf\implies\:2^{(13n+3)-6n}$

$\sf\implies\:2^{7n+3}$

______________

Formula's :

$\sf1)y {}^{m} \times y {}^{n} = y {}^{m + n}$

$\sf2) \dfrac{y {}^{m} }{y {}^{n} } = y {}^{m - n}$

$\sf3)y {}^{0} = 1$

Answer 2

${\underline{\red{\bf{Use\:the\:following\:identities\:;-}}}}$

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

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

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

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

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$\Large{\underline{\bf{\color{cyan}CaLcUlAtIoN,}}}$

$\pink\bigstar\:\:\bf{\dfrac{2^{-n}\times{8^{2n\:+\:1}}\times{16^{2n}}}{4^{3n}}\:}$

$\longmapsto\:\:\bf{\dfrac{2^{-n}\times{\Big\{(4\times{2})^{2n\:+\:1}\Big\}}\times{\Big\{(4\times{4})^{2n}\Big\}}}{4^{3n}}\:}$

$\longmapsto\:\:\bf{\dfrac{2^{-n}\times{2^{2n\:+\:1}}\times{4^{2n\:+\:1}}\times{4^{2n}}\times{4^{2n}}}{4^{3n}}\:}$

$\longmapsto\:\:\bf{\Big(2^{-n}\times{2^{2n\:+\:1}}\Big)\times\Big(4^{2n\:+\:1}\times{4^{2n}}\times{4^{2n}}\times{4^{-3n}}\Big)\:}$

$\longmapsto\:\:\bf{2^{(-n\:+\:2n\:+\:1)}\times{4^{(2n\:+\:1\:+\:2n\:+\:2n\:-\:3n)}}\:}$

$\longmapsto\:\:\bf{2^{n\:+\:1}\times{4^{(6n\:-\:3n\:+\:1)}}\:}$

$\longmapsto\:\:\bf{2^{n\:+\:1}\times{4^{3n\:+\:1}}\:}$

$\longmapsto\:\:\bf{2^{n\:+\:1}\times{\Big\{(2\times{2})^{3n\:+\:1}\Big\}}\:}$

$\longmapsto\:\:\bf{2^{n\:+\:1}\times{2^{3n\:+\:1}}\times{2^{3n\:+\:1}}\:}$

$\longmapsto\:\:\bf{2^{(n\:+\:1\:+\:3n\:+\:1\:+\:3n\:+\:1)}\:}$

$\longmapsto\:\:\bf\green{2^{7n\:+\:3}\:}$