Hello. I knew that on the GRE, we only consider the positive value for an even order of root of a number. Though I got confused by the above-marked rule. So, would that absolute value rule only apply when we’re considering the values of variables, and not for numbers?
Thank you.
You can just look at the graph if the algebra doesn’t make much sense to you. There’s only one output when x = 16. If there were two outputs for x = 16, then it wouldn’t be a function. We need these to act as functions because otherwise, we would lose a bunch of nice properties.
For example, let’s say we allowed \sqrt[4]{16} to be both 2 and -2. With that in mind, what would be the value of \sqrt[4]{16} + \sqrt[4]{16}? If we act on a whim and use different definitions as we please, then we end up conceding \sqrt[4]{16} + \sqrt[4]{16} = 2 + 2 = \textcolor{red}{4} = 2 - 2 = \textcolor{red}{0}, which is clearly nonsense.
A last possible question you may have is: why choose the positive output and not the negative output? The answer to that would just be practicality and convenience. Choosing the negative root would break rules we already use.
If we adopted the hypothetical negative root convention, then we would have:
Clearly 6 = -6 is false, so this “breaks” our exponent rule.
