Thank you for your participation in answering this challenging question. But in order to answer this question, an important question must be examined first to determine the task of the answer. What do you think about it?
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Oh, and to answer your original question, since the 0th root of a number is identical (for Reals) to raising a number to the 1/0 power, and since dividing by 0 is never allowed, the 0th root of 4 is undefined.
And that's all fine, since even numerically you might want to think of the 0th root as the same as raising 4 to the ∞ power... which again, is undefined.
Note one more time that nothing ever equals infinity. An answer can be undefined, though of course there are plenty of undefined things. One way something can be undefined is for that thing to be infinitely large, which is the case here.
So if you wanted the concept of ∞ to be included, you could say "4^∞ is infinitely large."
But that STILL doesn't let you "4^(1/0) is infinitely large." Dividing by zero halts you in your tracks. You can only throw up your hands and say "4^(1/0) is undefined" or "the 0th root of 4 is undefined."
Neither! Nothing ever EQUALS undefined. Nothing ever EQUALS infinity, either, unless we are working in the Extended Reals.
Instead, we say "1/0 is undefined."
But it is not proper mathematical terminology to say "1/0 = undefined."