how to prove cubic root of 25 is irrational using mathematical induction?
I've been trying to do it for hours but can't get it, help plz guys :S
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Let's suppose $r$ is a rational such that $r^3 = 25$, then there exist $a, b\in \mathbb N$ coprimes such that $$ a^3 = 25\, b^3 \,. $$ The above relation implies the following chain $$ 5 \mid a^3 \implies 5 \mid a \implies 5^3 \mid 25\, b^3 \implies 5 \mid b^3 \implies 5 \mid b $$ but that contradicts the assumption that $a$ anb $b$ are relatively prime.