Question

Explain the difference between the following two expressions. (-1)^2/4 and (-1)^1/2

Answer 1

$\bf a^{\frac{{ n}}{{ m}}} \implies \sqrt[{ m}]{a^{ n}} \qquad \qquad \sqrt[{ m}]{a^{ n}}\implies a^{\frac{{ n}}{{ m}}}\\\\ -------------------------------\\\\ (-1)^{\frac{2}{4}}\implies \sqrt[4]{(-1)^2}\implies \sqrt[4]{(-1)(-1)}\implies \sqrt[4]{1}\qquad \boxed{1^4=1} \\\\\\ \sqrt[4]{1^4}\implies 1\\\\ -------------------------------\\\\ (-1)^{\frac{1}{2}}\implies \sqrt[2]{(-1)^1}\implies \sqrt{-1}\implies i$