Why does e^1/8 = 1/x but when 1/x changes to x it = e^-1/8?

Because e^ (1/8) is the same as e^ (1/8) out of 1, when you flip 1 and x you also flip e^ (1/8) and 1, so it becomes 1 out of e^ (1/8). when the power is in the demoninator (the bottom) and you want to make it the numerator (move it to the top) you add a negative sign to the power. for example, 1 out of 2 squared if the same as 2 to the negative second power