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

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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