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Difference between revisions of "2021 JMPSC Sprint Problems/Problem 12"

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#[[JMPSC Problems and Solutions|All JMPSC Problems and Solutions]]
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{{JMPSC Notice}}

Revision as of 16:15, 11 July 2021

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


See also

  1. Other 2021 JMPSC Sprint Problems
  2. 2021 JMPSC Sprint Answer Key
  3. All JMPSC Problems and Solutions

The problems on this page are copyrighted by the Junior Mathematicians' Problem Solving Competition. JMPSC.png

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