Question

prove that 3 root under 5 is irrational, give that root under 5 is irrational.​

Answer 1

$\huge\fbox\colorbox{purple}{Solution}$

Given : 3 + √5

To Prove : 3 + √5 is an irrational number.

Proof :

Let us assume that 3 + √5 is a rational number.

So, it can be written in the form a/b

3 + √5 = a/b

Here a and b are coprime numbers and b ≠ 0

Solving 3 + √5 = a/b we get,

=>√5 = a/b – 3

=>√5 = (a-3b)/b

=>√5 = (a-3b)/b

This shows (a-3b)/b is a rational number. But we know that √5 is an irrational number.

So, it contradicts our assumption. Our assumption of 3 + √5 is a rational number is incorrect.

3 + √5 is an irrational number!

Hence Proved.