Why does $e^{(-x)^8}$ = $e^{x^8}$

Question

can someone please explain to me why $e^{(-x)^8}$ = $e^{x^8}$ ?

Edit: OH CRAP I CAN'T TYPE... MY TITLE WAS WRONG. But my question is the same. When I typed out on desmos: $e^{(-x)^8}$, I get the same curve as $e^{(x)^8}$

Edit 2: OH MY GOD... I READ IT WRONG. IT'S $e^{(-x)^8}$

Answer 1

Why does $e^{-x} = e^{x}?$

It doesn't, except for when $x = 0.$

---

Post-edit second question:

Why does $e^{(-x)^8} = e^{x^8}$?

This is, in fact, true. When you raise a negative to an even power, you get a positive. You may already know that $$(-1)^2 = (-1)\cdot (-1) = 1.$$

More generally, $$(-x)^{2n} = (-x)^n\cdot (-x)^n = (-1)\cdot(-1)\cdot x^n\cdot x^n = x^{2n}.$$

Try instead $e^{(-x)^7}$, which won't equal $e^{x^7}$ because $7$ isn't even.