Exponents under the radical confusion

[HughJanus]

In the problem below, I know that the inequality is ‘False’, because the answer is 3 < 0, but I’m slightly unclear on the concept here.

Specifically, what is the reason we can’t have the radical cancel out the exponent? Then we would be left with -3, and the inequality would be true.

So what is the rule that says we take care of the exponent under the radical first? Just slightly confused on like the order in which we do this. Cause in other problems (and maybe I’m mixing up concepts here) we kind of write something like this as ((-3)^2)^1/2 which would become (-3)^1 which would be -3. Or is that process not applicable here?

I feel like I’m missing something in my understanding of this concept.

[Leaderboard]

In general, \sqrt{x^2} = |x|, and you’re right in that \sqrt{(-3)^2} = 3,

@HughJanus

kind of write something like this as ((-3)^2)^1/2 which would become (-3)^1

The problem is that \sqrt{9} technically has two values, 3 and -3. However, on the GRE, only the positive value can be used.

[HughJanus]

Ok that makes sense. Thank you. So the method of used above to get to -3, is that not recommended since on the GRE the answer wouldn’t be -3 in this case? I’m just confused on when to use that method, cause I know that can be useful at times. (I’m talking about the method to remove the radical like writing it as ((-3)^2)^1/2)

Can you give an example of when that method is good to use and we wouldn’t run the risk of getting the wrong answer cause of that GRE rule?