If x^x= 22, which of the following could be a value of x ?

If x^x= 22, which of the following could be a… | GMAT Sprint

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PS · Problem Solving

705-805

Roots

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

Correct answer marked below

\frac{1}{16}

Correct

\frac{1}{4}

\frac{1}{\sqrt{2}}

\frac{1}{2}

\sqrt{2}

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Explanation

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

Take the above expression to the fourth power and apply exponentiation to the base

\frac{1}{2}

\left(\left(\frac{1}{2}\right)^{\frac{1}{4}}\right)^4 = \left(\left(\frac{1}{2}\right)^4\right)^{\frac{1}{4}} = \left(\frac{1}{16}\right)^{\frac{1}{4}}

Now, to compensate the previous operation, take the fourth root, but this time apply exponentiation to the exponent

\frac{1}{4}

:…

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What this question tests

Your handling of square and higher roots, including simplifying radicals.

Forgetting the ± when taking a square root to solve an equation, or assuming √(a²) = a.