the only integer n for which n equals x to the y power equals y to the x power for x does not equal y
4^2 = 2^4
thanks!!
2^4 = 16 = 4^2
To find the only integer solution for the equation x^y = y^x, where x ≠ y, you can approach it by observing the properties of the equation.
If x = y, then the equation becomes x^x = x^x, which is always true for any positive integer x.
So, let's consider the case where x ≠ y.
We can rearrange the equation x^y = y^x as follows:
x^(1/x) = y^(1/y)
Now, we know that for any positive integer x, x^(1/x) is a decreasing function. Similarly, y^(1/y) is also a decreasing function.
Since x ≠ y, the two expressions x^(1/x) and y^(1/y) will cross each other at one and only one point.
Hence, we need to find the point of intersection of the two expressions to get the solution to the equation.
To do this, you can plot the graphs of x^(1/x) and y^(1/y) for different values of x and y.
When you plot the graphs, you will find that they intersect at the point (4, 2).
Therefore, the only integer solution to the equation x^y = y^x, where x ≠ y, is when x = 4 and y = 2.
In other words, 4^2 = 2^4.