Why Is \sqrt[4]{(-4)^2} Not Equal to (-4)^\frac{1}{2}?

[bacon]

From the book..." $$\sqrt[4]{(-4)^2}$$=$$\sqrt[4]{16}$$=2. It is incorrect to write $$\sqrt[4]{(-4)^2}$$=$$(-4)}^\frac{2}{4}$$=$$(-4)}^\frac{1}{2}$$=$$\sqrt{-4}$$ ..."

I understand the math involved but want to be sure of the exact reason why the first part is correct and the second is not. Is it because of the inner to outer priority of operations when one operation is nested inside another?

Thanks for any answers.

 

[rocomath]

Work inner ... outer.

 

[CompuChip]

Clearly, the first method is correct (it actually says $((-4)^2)^{1/4}$, so what it does is work out the brackets in the correct order.
Now if the second method were correct, you would get contradictory results. For example, consider this "proof":
$$1 = \sqrt{1} = \sqrt{(-1)^2} = ((-1)^2)^{1/2} \stackrel{?!}{=} (-1)^{2/2} = (-1)^1 = -1$$
so 1 = -1, and anything you might want to prove (whether true or false) follows

 

[Feldoh]

1 = -1

My life has been a lie :(

 

[bacon]

My life has been a lie :(

Actually, it is not. I could show you a proof of this, but I need to change the oil in the car. Sorry.

 

[CompuChip]

My life has been a lie :(

It's not that bad... have some cake.