Question

Prove that 3 root 5-2 is an irrational number

Answer 1

Answer:

3√5 - 2

Let us assume, to the contrary, that 3√5 - 2 is rational, such that it can be expressed in the form a/b, where a and b are integers and co-prime.

3√5 - 2 = $\dfrac{a}{b}$

⇒ 3√5 = $\dfrac{a}{b}$ + 2

⇒ 3√5 = $\dfrac{a+2b}{b}$

⇒ √5 = $\dfrac{a+2b}{3b}$

Since, a and b are integers, therefore $\frac{a+2b}{3b}$ must be rational. So, √5 is also rational.

But this contradicts the fact that √5 is irrational.

Therefore, 3√5 - 2 is irrational.