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

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Note that <math>\sqrt{x}=x^{\frac{1}{2}}</math>. So <math>{x}\sqrt{x^{\frac{3}{2}}}=2^{63}</math>. Simplifying gives that <math>x{\cdot}x^{\frac{3}{4}}=x^{\frac{7}{4}}=2^{63}</math>. If <math>x</math> is <math>2^m</math>, then <math>\frac{7m}{4}=63</math>, so <math>m=\boxed{36}</math>.
 
Note that <math>\sqrt{x}=x^{\frac{1}{2}}</math>. So <math>{x}\sqrt{x^{\frac{3}{2}}}=2^{63}</math>. Simplifying gives that <math>x{\cdot}x^{\frac{3}{4}}=x^{\frac{7}{4}}=2^{63}</math>. If <math>x</math> is <math>2^m</math>, then <math>\frac{7m}{4}=63</math>, so <math>m=\boxed{36}</math>.
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==Solution 3==
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We square both sides of the equation to get <cmath>x^2 \cdot x\sqrt{x}=x^2\sqrt{x}=2^{63 \cdot 2}.</cmath> We square both sides of the equation again to get <cmath>x^6 \cdot x=x^7=2^{63 \cdot 4}.</cmath> Thus, <math>x=2^{63 \cdot 4/7}=2^{36}</math>, so the answer is <math>\boxed{36}</math>.

Revision as of 11:51, 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 \[x^2 \cdot x\sqrt{x}=x^2\sqrt{x}=2^{63 \cdot 2}.\] 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 $\boxed{36}$.

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