Basic exponent question/review

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I have function $L^2=kL$

If I solve for L, I divide both sides by L and get L=K.

I now have another function $L^A=kL$. To solve for L, I raise both sides to 1/A, and get $L=\left(KL\right)^{\frac{1}{A}}$.

From the first function, A=2. If I plug A=2 into the second function, I don't get the same answer.

Question: I don't understand why in the first function I can't raise both sides to 1/2, but in the second function I CAN raise both sides to 1/A.

Thanks in advance!