Question

find the value of (256) -3/4

Answer 1

$\sf{Solution:-}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \:$

$\sf\purple{First \: Find \: Prime \: Factorisation \: of \: 256}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \:$

$\begin{array}{r | l}2&256\\2&128\\2&64\\2&32\\2&16\\2&8\\2&4\\2&2\\&1\end{array}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \:$

$\purple\leadsto\sf{2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2}$

$\sf\purple{Make \: Pairs \: of \: 2}$

$\purple\leadsto\sf{ \underbrace{2 \times 2} \times \underbrace{2 \times 2} \times \underbrace { 2 \times 2 }\times \underbrace{2 \times 2}}$

$\purple\leadsto\sf{ 2 \times 2 \times 2 \times 2}$

$\purple\leadsto\sf{ {2}^{4} }$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \:$

$\sf \purple{Now \: Solve \: Ques}$

$\purple\leadsto\sf{ {256}^{ \frac{ - 3}{4} } }$

$\sf\purple{We \: also \: write \: 256 \: as \: }\sf{ {2}^{4} }$

$\purple\leadsto\sf{ ( { {2}^{4}})^{ \frac{ - 3}{4} } }$

$\purple\leadsto\sf{ {2}^{ \cancel4 \times \frac{ - 3}{ \cancel4} } }$

$\purple\leadsto\sf{ {2}^{ - 3} }$

$\purple\leadsto\sf{ {a}^{ - m} = \frac{1}{ {a}^{m} } }$

$\purple\leadsto\sf{ { \frac{1}{(2)^{3} } } }$

$\purple\leadsto\sf{ \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} }$

$\purple\leadsto\sf{ \frac{1}{8} }$