Question

If we are given x such that 25^x-9^y=18 and 5^x-3^y=3, compute 5^x+3^y.

Answer 1

25 = 5*5 and 9 = 3*3, which we can exploit to write

`25^x-9^y=(5^2)^x-(3^2)^y=5^(2x)-3^(2y)=(5^x)^2-(3^y)^2`

so that this expression is actually a difference of squares. We can factorize this to get

`(5^x-3^y)(5^x+3^y)=18`

and given that
`5^x-3^y=3`, we divide both sides by this to get

`5^x+3^y=\frac{18}3=\boxed{6}`