If x^x=(2^(1/2)/2)^(1/2), which of the following could be a value of x

Official Solution:

If \(x^x=\sqrt{\frac{\sqrt{2}}{2}}\), which of the following could be a value of \(x\)?

A. \(\frac{1}{16}\)
B. \(\frac{1}{4}\)
C. \(\frac{1}{\sqrt{2}}\)
D. \(\frac{1}{2}\)
E. \(\sqrt{2}\)


We need to simplify the expression on the right-hand side of the equation so that it can be written in terms of the same base and exponent as \(x^x\).

\(\sqrt{\frac{\sqrt{2}}{2}}=\)

\(=\sqrt{\frac{1}{\sqrt{2}}}=\)

\(=\sqrt{(\frac{1}{2})^{\frac{1}{2}}}=\)

\(=(\frac{1}{2})^{\frac{1}{4}}=\)

Take to the above expression the fourth power and apply exponentiation to the base, \(\frac{1}{2}\):

\(=((\frac{1}{2})^{\frac{1}{4}})^4=\)

\(=((\frac{1}{2})^4)^{\frac{1}{4}}=\)

\(=(\frac{1}{16})^{\frac{1}{4}}\)

Now, to compensate the previous operation, take the fourth root, but this time apply exponentiation to the exponent, \(\frac{1}{4}\):

\(=((\frac{1}{16})^{\frac{1}{4}})^{\frac{1}{4}}=\)

\(=(\frac{1}{16})^{\frac{1}{16}}\)

Therefor, \(x=\frac{1}{16}\)

Else, after obtaining the expression \(x^x=(\frac{1}{2})^{\frac{1}{4}}\), we can substitute each of the given options for \(x\) to determine which one satisfies the equation.

Option A is a valid solution, as \((\frac{1}{16})^{(\frac{1}{16})}=((\frac{1}{16})^{(\frac{1}{4})})^{(\frac{1}{4})}=(\frac{1}{2})^{\frac{1}{4}}\)


Answer: A