Question

S.T cube root of 2 is irrational​

Answer 1

Let us assume that √2 is a rational number.

So, 2=a/b(b≠0), where a and b are co-primes.

Taking square on both the sides,

√2²=a²/b²

=2=b²|a²

=2b=a² ------------------(1)

This means 2 divides a² and so 2 divides a.

Let a=c, where c is another integer.

a=2c

Taking square on both sides,

a²=4c² -------------------(2)

from (1) and (2)

2b²=4c²

=b²=4c²/2

=b²=2c²

This means 2 divides b² and so 2 divides b.

This means that a and b have at least 2 and 1 as common factor. Hence, they aren't co-primes. But this contradicts the fact that a and b are co-primes. This contradiction occurred because we assumed √2 as rational. Hence, our assumption is wrong. Therefore, √2 is an irrational number.

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