Expression will become 512^(2*1/3*1/2*1/3)

So we are left with (512)^(1/3*1/3)

cube root of 512 is 8. So we are now left with 8^1/3 which is 2.

Answer is option D.

Bunuel wrote:
What is the value of \((\sqrt[3]{\sqrt{\sqrt[3]{512}}})^2\)?

A. 1/16

B. 1/8

C. 1/2

D. 2

E. 8

\((\sqrt[3]{\sqrt{\sqrt[3]{512}}})^2\) -------(cube root of 512 = 8)
= \((\sqrt[3]{\sqrt8})^2\) ------- (squaring \((\sqrt{8})^2\) we get 8 )
= \((\sqrt[3]{8})\)
= \((\sqrt[3]{8})\) = 2
Answer D...

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512 = 8*8*8..

hence solving the expression gives as 2

ans D

\(\Bigg (\sqrt[3]{\sqrt{\sqrt[3]{512}}}\Bigg )^2=\bigg (\sqrt[3]{\sqrt{8}} \bigg )^2 \\
= \bigg (\sqrt[3]{\sqrt{2^3}} \bigg )^2=\big (\sqrt{2} \big )^2=2\)

Or we could solve in this way:

\(\Bigg (\sqrt[3]{\sqrt{\sqrt[3]{512}}}\Bigg )^2 =\Bigg (\sqrt[3]{\sqrt{512^{\frac{1}{3}}}}\Bigg )^2 \\
= \Bigg (\sqrt[3]{\Big ( 512^{\frac{1}{3}} \Big )^{\frac{1}{2}}}\Bigg )^2 = \Bigg (\sqrt[3]{ 512^{\frac{1}{6}} }\Bigg )^2 \\
= \Bigg ( \Big (512^{\frac{1}{6}} \Big ) ^ {\frac{1}{3}}\Bigg )^2 = \Bigg ( 512^{\frac{1}{18}}\Bigg )^2 \\
= 512^{\frac{1}{9}} = (2^9)^{\frac{1}{9}} = 2\)
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Every cube root can be replaced by 1/3 and square root can be replaced by 1/2 in the power.

\(512 = 2^9\)

\((\sqrt[3]{\sqrt{\sqrt[3]{512}}})^2\)

\(= (2^9)^{2*(1/3)*(1/2)*(1/3)}\)

\(= 2\)
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\((\sqrt[3]{\sqrt{\sqrt[3]{512}}})^2\)
= (512)^(2*1/3*1/2*1/3)
= (512)^1/9
= 2

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