2021 JMPSC Sprint Problems/Problem 12

Problem

The solution to the equation $x \sqrt{x \sqrt{x}}=2^{63}$ can be written as $2^m$ , where $m$ is a real number. What is $m$ ?

Solution

Let $y=\sqrt[4]{x}.$ Then, we have that the expression on the left hand side is equivalent to $y^4\cdot y^2\cdot y=y^7.$ Thus, we have that $y^7=2^{63}.$ Taking the 7th root of both sides gives $y=2^9,$ thus we have $\sqrt[4]{x}=2^9,$ which makes $x=2^{36}.$ Answer is $\boxed{36}.$

~Lamboreghini

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2021 JMPSC Sprint Problems/Problem 12

Problem

Solution