# 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 $y=\sqrt{x}.$ 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 $\sqrt{x}=2^9,$ which makes $x=2^{36}.$ Answer is $\boxed{36}.$

~Lamboreghini

## Solution 2

Note that $\sqrt{x}=x^{\frac{1}{2}}$. So ${x}\sqrt{x^{\frac{3}{2}}}=2^{63}$. Simplifying gives that $x{\cdot}x^{\frac{3}{4}}=x^{\frac{7}{4}}=2^{63}$. If $x$ is $2^m$, then $\frac{7m}{4}=63$, so $m=\boxed{36}$.

## Solution 3

We square both sides of the equation to get $$x^2 \cdot x\sqrt{x}=x^3\sqrt{x}=2^{63 \cdot 2}.$$ We square both sides of the equation again to get $$x^6 \cdot x=x^7=2^{63 \cdot 4}.$$ Thus, $x=2^{63 \cdot 4/7}=2^{36}$, so the answer is $\boxed{36}$.

~tigerzhang

## Solution 4

We can divide both sides by $x$ to get $\frac{2^{63}}{x} = \sqrt{x \sqrt{x}}$. Squaring both sides gives $\frac{2^{126}}{x^2} = x \sqrt{x}$. Dividing both sides by $x$ gives $\frac{2^{126}}{x^3} = \sqrt{x}$. Squaring both sides again gives $\frac{2^{252}}{x^6} = x$. Dividing both sides gives $\frac{2^{252}}{x^7} = 1$. We can factor this as $\left(\frac{2^{36}}{x}\right)^7 = 1$. We know that since $m$ is a real number, $2^m$ also must be real, and since $2^m$ is real, $x$ must be real. We can take the 7th root on both sides to get $\frac{2^{36}}{x} = 1$. Multiplying both sides by $x$ gives $2^{36} = x$. We know that $2^m = 2^{36}$, which means that $m = \boxed{36}$.

~hh99754539

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