Easy Root

[rocdocmac]

Find x in:

 

 

[bonanova]

The solutions are

Spoiler

1 and 2.

 

Because
 

Spoiler

x ^1/x = (x²)^{1/x^2} = x ^{2/x^2}

By inspection 1 and 2 are solutions, and a plot of the difference of x ^1/x and x ^{2/x^x} shows sign changes (only) at 1 and 2.

Also, for any value of x not equal to 0 (illegal division) or 1 (all powers of 1 are equal) dividing by x yields simply

1/x = 2/x² or more simply 

x² = 2x.

Cute.

[rocdocmac]

Spoiler

Indeed a very easy one! I was first thinking of having one find x and y in:

"xth root of x = yth root of y" [or x^(1/x) = y^(1/y)]

If x<>y, then only one solution:

x = 2 and y = 4