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Question:

The simplest value of the expression ${\{\frac{4^{p + \frac{1}{4}} \times \sqrt{2 \times 2^{p}}}{2 \times \sqrt{2^{- p}}}\}}^{\frac{1}{p}}$ is:

Explanation:

$\{\frac{4^{p + \frac{1}{4}} \times \sqrt{2 \times 2^{p}}}{2 \times \sqrt{2^{- p}}}\} = \{\frac{4^{p} \times 4^{\frac{1}{4}} \times \sqrt{2} \times \sqrt{2^{p}}}{2 \times \sqrt{2^{- p}}}\} = \{\frac{2^{2 p} \times \sqrt{2} \times \sqrt{2} \times \sqrt{2^{p}}}{2 \times \sqrt{2^{- p}}}\} = 2^{2 p} \times \sqrt{2^{p}} \times \sqrt{2^{p}} = 2^{3 p}$

${\{\frac{4^{p + \frac{1}{4}} \times \sqrt{2 \times 2^{p}}}{2 \times \sqrt{2^{- p}}}\}}^{\frac{1}{p}} = {(2^{3 p})}^{\frac{1}{p}} = 8$

Hence, option (b).

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