why do I invert the exponent (Economics)

Question

This question refers to the Capital stock in macroeconomics, so K is the capital stock.

Hi everyone. I have this problem where I don't actually understand why you invert the exponent. You still have to remove the exponent in the 4th step so why invert it in the first place. Is this because of the negative exponent it will give because of alpha being so low?

Answer 1

They want to go from the second line to fifth line (and then probably to $K = \dots$) and are just taking baby steps:

• 2 to 3: invert both sides;
• 3 to 4: simplify both sides;
• 4 to 5: raise both sides to the power $1/(1-\alpha)$.

They might as well have taken all those steps as one; the authors probably believe this is easier to understand for their readers.