Question

How can the properties of rational exponents be applied to simplify expressions with radicals or rational exponents?

Answer 1

`\bf a^{\frac{{ n}}{{ m}}} \implies \sqrt[{ m}]{a^( n)} \\\\\\ \sqrt[{ m}]{a^( n)}\implies a^{\frac{{ n}}{{ m}}}\\\\ -----------------------------\\\\ \textit{let's say you have }\qquad \sqrt[27]{64^9} \\\\ \sqrt[27]{64^9}\implies 64^{\cfrac{}{}(9)/(27)}\quad \quad however\implies 64=4^3 \\\\ thus\qquad 64^{\cfrac{}{}(9)/(27)}\implies (4^3)^{\cfrac{}{}(9)/(27)}\implies (4)^{\cfrac{}{}3\cdot (9)/(27)}\implies 4^{\cfrac{}{}(9)/(9)}\implies 4^1\implies 4`