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

$5^x+3^y=\dfrac{18}3=\boxed{6}$