Question

Prove that ³√2 is an irrational number.

Answer 1

hello there!!
if possible ,let ³√2is rational .then there exist positive co prime a and b such that

³√2=a/b
=>2=a³/b³[taking cube on both sides]

=> 2b³=a³------(1)

=> 2divides a³[since 2divides 2b³]
2 is a prime which divides a³ so it will divide a also
now let a =2c for some integer c
put in (1)

2b³=(2c)³
=> 2b³=8c³
=>b³=4c³
2divides b³ [since 2 divides 4c³]
2 is prime which divides b³ so it will divide b also
but this contradicts the fact that a and b are co primes
hence,³√2 is an irrational number .