Evaluate the experssion of Power of Power

Question:

$\sqrt[3]{4}\sqrt[3]{\frac{1}{500}}$

Solution:

Remember:

$\frac{\sqrt[a]{b}}{{\sqrt[a]{c}}}= \sqrt[a]\frac{b}{c}$

So, 

  $\sqrt[3]{4}.\sqrt[3]{\frac{1}{500}}=\sqrt[3]{{4}.\frac{1}{500}}$

  $=\sqrt[3]{2^{2}.\frac{1}{2^{2}.5^{3}}}$          [$500 = 2^{2}.5^{3}, 4=2^{2}$]

  $=\sqrt[3]{\frac{1}{5^{3}}}$

  $= {\frac{1}{5}}^{\frac{3}{3}}$                         [Recall, $\sqrt[a]{b^{m}}= b^{\frac{m}{a}}$]

  $= {\frac{1}{5}}^{1}$

  $= \frac{1}{5}$