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Given that sum of fifth root of (1/64 + x) and (1/64 - x) is equals to 1/2. Then find the value of x.
Given equation is,
\[\sqrt[5]{\frac{1}{64}+x}+\sqrt[5]{\frac{1}{64}-x}=\frac{1}{2}\]
Let,
\[a=\frac{1}{64}+x\]
\[b=\frac{1}{64}-x\]
Observe that,
\[a+b=\frac{1}{32}=\left(\frac{1}{2}\right)^5----(1)\]
From given equation,
\[\sqrt[5]{a}+\sqrt[5]{b}=\frac{1}{2}----(2)\]
From (1) and (2):
a = 0 or b = 0;
\[\Rightarrow x=\pm\frac{1}{32}=\pm\left(\frac{1}{2}\right)^5\]
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