Question

INDUCTION Show that (ab)^n=(a^n)(b^n)

Answer 1

for \(n=1\) we have \((ab)^1=ab=(a^1)(b^1)\) which is trivially true.
assume it holds for some \(k\) i.e. \((ab)^k=(a^k)(b^k)\). now observe what happens by multiplying both sides by \(ab\):$$(ab)^k=(a^k)(b^k)\\(ab)^k(ab)=(a^k)(b^k)(ab)\\(ab)^{k+1}=(a^ka)(b^kb)=(a^{k+1})(b^{k+1})$$

Answer 2

every time I think of induction I remember my crazy Chinese teacher that went over it in number theory

YOU IN FRONT OF ME
I AM GOING
YOU MUST GO
then he yelled GO GO GO !