Question

Prove that 2root 3 plus root 5 is irrational

Answer 1

Answer:

2√3 + 5

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

2√3 + 5 = $\dfrac{a}{b}$

⇒ 2√3 = $\dfrac{a}{b}$ - 5

⇒ 2√3 = $\dfrac{a-5b}{b}$

⇒ √3 = $\dfrac{a-5b}{2b}$

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

But this contradicts the fact that √3 is irrational.

Therefore, 2√3 + 5 is irrational.