Question

1. Prove that 3 +2 √5 is irrational​

Answer 1

Question :

Prove that 3+2√5 is irrational.

Solution:

Let us Assume that 3+2√5 is rational.

Thus it can be written in the form of

$\frac{a}{b} \: and \: a \: and \: b \: are \: coprime.$

Therefore

$3 + \sqrt{5} = \frac{a}{b} \\ \\ 3 - \frac{a}{b} = 2 \sqrt{5} \\ \\ \frac{3b - a}{b} = \sqrt{5}$

Since a and b are integers then we get

$\frac{3b - a}{b}$

is rational number and √5 is also rational number.

But it is not true because √5 is irrational.

Therefore the given number 3+√5 is irrational and our assumption taken is wrong.