Products of e to fractional exponents

Question

I'm quite happy dealing with simple exponents, or so I thought. I've come across the following product and solution:

$$ e^{-x^2/2\alpha^2}\times e^{-2x^2/\alpha^2}=e^{-5x^2/2\alpha^2} $$

When I work this through, I get the following instead:

$$ e^{\frac{-x^2-2x^2}{2\alpha^2+\alpha^2}} \rightarrow e^{-3x^2/3\alpha^2} $$

I presumed I would just use addition on the exponents as they are the same base. But the 2 coefficient threw me. Should I look to simplify them first?

Answer 1

You have made a mistake when adding the fractions: the exponent should be $$-\frac{x^2}{2\alpha^2}-\frac{2x^2}{\alpha^2}=-\frac{x^2}{2\alpha^2}-\frac{4x^2}{2\alpha^2}=-\frac{5x^2}{2\alpha^2}$$

Answer 2

Since $-\frac{x^2}{2 \alpha ^2} -\frac{2 x^2}{\alpha ^2} = -\frac{5 x^2}{2 \alpha ^2} \neq \frac{-3 x^2}{3 \alpha ^2} = \frac{-x^2 - 2 x^2}{2 \alpha ^2 + \alpha ^2} $, the first formula is correct.

Answer 3

The problem is the addition of the demoninator. You are right that $$e^{-x^2/2a^2}\cdot e^{-2x^2/a^2}=e^{-x^2/2a^2-4x^2/a^2}=e^{-5x^2/a^2}$$