Difference between revisions of "2021 JMPSC Sprint Problems/Problem 12"
Lamboreghini (talk | contribs) (→Solution) |
(→Solution) |
||
Line 6: | Line 6: | ||
~Lamboreghini | ~Lamboreghini | ||
+ | |||
+ | ==Solution 2== | ||
+ | |||
+ | 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>. |
Revision as of 07:39, 11 July 2021
Problem
The solution to the equation can be written as
, where
is a real number. What is
?
Solution
Let 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
which makes
Answer is
~Lamboreghini
Solution 2
Note that . So
. Simplifying gives that
. If
is
, then
, so
.