Question

Prove that 3 root 6 is not a rational number

Answer 1

let 3√6 be rational. an equal to a/b. where a,b≠0 are rational.. then a/b must be rational. but this contradicts that 3√6 is rational. hence 3√6 is irrational

Answer 2

Answer:

Suppose 3√6 is a rational number,

Thus, by the property of rational number,

We can write,

$3\sqrt{6}=\frac{p}{q}$

Where, p and q are distinct  integers and q ≠ 0,

$3\sqrt{6}q=p$

By squaring both sides,

$54q^2=p^2$ ------(1)

⇒ p² is a multiple of 54,

⇒ p is a multiple of 54

Thus, we can write,

p = 54a, where a is any number,

From equation (1),

$54q^2=2916a^2$

$q^2=54a^2$

⇒ q² is a multiple of 54,

⇒ q is a multiple of 54

Therefore, p and q are not distinct,

Which is a contradiction,

⇒ 3√6 is not a rational number,

Thus, 3√6 is an irrational number

Hence, proved......