Evaluate the experssion of Power of Power

Question:

a)

$\sqrt[5]{\frac{1}{8}}\sqrt[5]{\frac{1}{4}}$

Solution:

Remember, $\sqrt[a]{c}\sqrt[a]{d} = \sqrt[a]{cd}$

So, 

  $\sqrt[5]{\frac{1}{8}}\sqrt[5]{\frac{1}{4}} =\sqrt[5]{{\frac{1}{8}}.{\frac{1}{4}}}$

  $= \sqrt[5]{\frac{1}{8.4}}$

  $= \sqrt[5]{\frac{1}{32}}$

  We can simplify it further,

  32 = 2.2.2.2.2 = $2^{5}$

  So, 

  $= \sqrt[5]{\frac{1}{2^{5}}}$            [Recall, $\sqrt[a]{b^{m}}= b^{\frac{m}{a}}$]

  $= {(\frac{1}{2})}^{\frac{5}{5}}$

  $= {(\frac{1}{2})}^{1}$

  $= \frac{1}{2}$