Question

Prove that √3 +2√5 is a irrational number.​

Answer 1

Step-by-step explanation:

$\text{First I'll prove that \sqrt3 is irrational.}\\ \text{Let is suppose that \sqrt3 is rational}\\ \text{ So, it can be expressed in the form of \sqrt{a}{b} where a and b have no other factor than 1\\ \\ \implies b\sqrt3=a [cross multiply] \\ \\ squaring both sides\\ 3b^2=a^2 \\ \therefore by division lemma, 3 and b^2 divides a^2 \\ \therefore 3 divides a}$$\text Now, let a=3c\\ the equation will be 3b^2=9c^2\\ So, b^2=3c^2\\ \thereofore 3 and c^2 divides b^2. So, 3 divided b\\ But we know that, a and b have no other factors other than 1. This contradiction arises because we considered \sqrt3 as rational. \therefore \sqrt3 is irrational.}$