# 2021 JMPSC Sprint Problems/Problem 12

## 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 .

## Solution 3

We square both sides of the equation to get We square both sides of the equation again to get Thus, , so the answer is .

~tigerzhang

## Solution 4

We can divide both sides by to get . Squaring both sides gives . Dividing both sides by gives . Squaring both sides again gives . Dividing both sides gives . We can factor this as . We know that since is a real number, also must be real, and since is real, must be real. We can take the 7th root on both sides to get . Multiplying both sides by gives . We know that , which means that .

~hh99754539

## See also

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