Difference between revisions of "2021 JMPSC Sprint Problems/Problem 12"
(Created page with "==Problem== The solution to the equation <math>x \sqrt{x \sqrt{x}}=2^{63}</math> can be written as <math>2^m</math>, where <math>m</math> is a real number. What is <math>m</ma...") |
Lamboreghini (talk | contribs) (→Solution) |
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==Solution== | ==Solution== | ||
| − | + | Let <math>y=\sqrt[4]{x}.</math> Then, we have that the expression on the left hand side is equivalent to <math>y^4\cdot y^2\cdot y=y^7.</math> Thus, we have that <math>y^7=2^{63}.</math> Taking the 7th root of both sides gives <math>y=2^9,</math> thus we have <math>\sqrt[4]{x}=2^9,</math> which makes <math>x=2^{36}.</math> Answer is <math>\boxed{36}.</math> | |
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| + | ~Lamboreghini | ||
Revision as of 23:05, 10 July 2021
Problem
The solution to the equation 
can be written as 
, where 
is a real number. What is ?
Solution
Let ![$y=\sqrt[4]{x}.$](http://latex.artofproblemsolving.com/f/c/7/fc7f3c4756b1093d72e06046f4a812aa14a940fa.png)
Then, we have that the expression on the left hand side is equivalent to 
Thus, we have that 
Taking the 7th root of both sides gives 
thus we have ![$\sqrt[4]{x}=2^9,$](http://latex.artofproblemsolving.com/f/8/7/f8755d735ab21afd48295867bf80dcfa2544a09a.png)
which makes 
Answer is 
~Lamboreghini
