0 Raised to 0 | Infinite versus Non-Defined

Thanks, it helped. I do not find any methof to prove 0^0 as 1 (however I can prove 0^0 as undefined assuming 0 divided by 0 is undefined) - anyway as you suggests and I take it as a word of caution that this won't appear in GMAT so I need not to worry. Cool

Sorry, just noticed this post.

Thanks Bunuel, for answering this on my behalf.

Kudos!

Yes, logically 0^0 is not defined. Once again, the reasoning is-

'Non defined' suggests that a unique, defined answer is not possible.
Examples:
0/0 (0 divided by anything is 0 but anything divided by 0 is infinite; hence, 'non-defined').
0^0 (0 raised to power anything is 0 but anything raised to power 0 is 1; hence, 'non-defined').

Nonetheless, as Bunuel mentioned, this won't appear on GMAT.

All the best!
Experts' Global

This is not correct. 1/0 and 2/0 are undefined, just as 0^0 is undefined. 1/0 and 2/0 are not "infinite". Any mathematical reference at all will confirm this.

One way to see why this is the case: for 0^0, say, if we take the expression 0^x, where x is positive, we know 0^x = 0. So if we imagine making x infinitely close to 0, 0^x should be 0. But if we do the same for x^0, the answer should be 1. There's no way to decide whether 0^0 should be 0 or 1, so it is undefined. The same is true for 1/x. If we imagine x getting infinitely close to 0, from above, i.e. when x is positive, 1/x gets larger and larger. But if we imagine x approaching zero from below (so x is negative), 1/x gets smaller and smaller. So it does not make sense to say 1/0 is "infinite", since it makes as much sense to say it is "negatively infinite".

These are really calculus questions, so they're out of the scope of the GMAT, but for GMAT purposes, test takers should consider x/0 to be undefined (or in other words "mathematically nonsensical"). Everything Bunuel says above about 0 is correct, though the external link he cites about 0^0 is not correct; that expression is undefined on the GMAT, and there will absolutely never be a circumstance where it "appears in some format or is a required intermediary calculation".

Nicely put. However, if I am gonna prove 0^0 as undefined via equations it would be as following:
0^0 = 0^(1-1) {1-1=0; hence putting 0 as 1-1}
0^(1-1) = (0^1 * 0^-1) = (0^1/0^1) (using power operations)
hence 0^0 = 0/0 and which is undefined.
Again, not relevant to GMAT