# Algebraic Proof of Sum of Exponential Powers is Product of Exponentials

Can somebody provide a proof of the summation of powers law for the product of two exponentials, using only algebra and the Taylor series, no derivatives or calculus tricks?

• You mean proving the functional equation $f(x)f(y) = f(x+y)$?
– mvw
Jul 10, 2014 at 19:56
• ... but the derivative tricks are much more fun Jul 10, 2014 at 20:33
• ^ Yes and yes. I suppose it's not that bad, I thought it was worse.
– user82004
Jul 10, 2014 at 21:01

\begin{align} \exp{(a+b)}&=\sum_{n=0}^{\infty}\frac{(a+b)^n}{n!}\\ &=\sum_{n=0}^{\infty}\sum_{k=0}^{n}\binom{n}{k}\frac{a^kb^{n-k}}{n!}\\ &=\sum_{n=0}^{\infty}\sum_{k=0}^{n}\frac{a^kb^{n-k}}{k!(n-k)!}\\ &=\left(\sum_{k=0}^{\infty}\frac{a^k}{k!}\right)\cdot\left(\sum_{n=0}^{\infty}\frac{b^n}{n!}\right)\\ &=\exp{(a)}\cdot\exp{(b)}.~~\blacksquare \end{align}