I think I have an error in my calculation, because I should get at the end but I do not. Can someone please check to see where I've gone wrong?

Suppose you have:

Rewrite as

Expand the brackets:

To find golden rule consumption, take the partial:

Set that beastie equal to zero so we can find the local max:

Divide through by to get

Multiply both sides by

Raise both sides to the power of

Multiply both sides by

And finish by raising both sides by

We can rewrite as

But that hardly makes it look closer to , which I think is the right answer. What have I done wrong?