3 ms·
Me, me! Let me try! X^N * X^M = N copies of X, multiplied by M copies of X = N+M copies of X multiplied together = X^(N+M)
by cousin_it 15y ago
Me, me! Let me try!
X^N * X^M
= N copies of X, multiplied by M copies of X
= N+M copies of X multiplied together
= X^(N+M)
- baddox 15y agoThat's not very rigorous, especially with fractional (or God forbid, irrational) exponents.
- cousin_it 15y agoYeah, and I'd be even more interested in a similarly intuitive explanation for the case of complex exponents ;-)
- baddox 15y agoThat would be difficult, since a formal construction of even the real numbers is a somewhat advanced (3rd or 4th year college mathematics) topic. I forget the details, but I believe a^n for real a and complex n is formally defined using the exponential function (e^x).
- jules 15y agoAt least in the Netherlands, construction of the reals (for example from rational Cauchy sequences) is standard 1st semester stuff. Understanding reals is required or provides a good source of examples for virtually all mathematics courses, so I can't imagine how some universities teach mathematics without it.
- jerf 15y agoGiven that the exam is from 1869 for entrance into college, I suspect that's basically what they are looking for.
- Queue29 15y agoYes, because irrational numbers wouldn't be discovered until the 1900's....
- bbrizzi 15y agoActually one of the ways to prove that requires the fact that R completes Q. The concept of completeness was introduced by Cauchy in the 19th century and it might not have been completely popularised in 1869.
- impendia 15y agoIndeed this is informal and could be made more rigorous, but even at the highest level of rigor, I think it's most natural to do integral exponents and then rational. Indeed you have to construct the integers before you can construct the rationals.