Question

What is the simplified value of ${\left\{\frac{4^{m+\frac{1}{4}}\times \sqrt{2\times 2^m}}{2\sqrt{2^{-m}}}\right\}}^{\frac{1}{m}}$ ?

Answer 1

${\left\{\frac{4^{m+\frac{1}{4}}\times \sqrt{2.2^m}}{2\sqrt{2^{-m}}}\right\}}^{\frac{1}{m}}={\left\{\frac{(2^2{)}^{m+\frac{1}{4}}\times (2^{m+1}{)}^{\frac{1}{2}}}{2\times 2^{\frac{-m}{2}}}\right\}}^{\frac{1}{m}}$

$={\left\{\frac{2^{2m+\frac{1}{2}}\times 2^{\frac{m}{2}+\frac{1}{2}}}{2^{1-\frac{m}{2}}}\right\}}^{\frac{1}{m}}$

$={\left\{2^{2m+\frac{1}{2}+\frac{m}{2}+\frac{1}{2}-1+\frac{m}{2}}\right\}}^{\frac{1}{m}}$

$=(2^{3m}{)}^{\frac{1}{m}}$

$=2^3=8$