Question

prove that the following are irrational root 3+root5​

Answer 1

To prove :

$\sqrt{3} + \sqrt{5}$

is irrational.

Let us assume it to be a rational number.

Rational numbers are the ones that can be expressed in

$\frac{p}{q}$

form where p,q are integers and q isn't equal to zero .

$\sqrt{3} + \sqrt{5} = \frac{p}{q} \\ \sqrt{3} + \frac{p}{q} - \sqrt{5}$

squaring on both sides,

As p and q are integers RHS is also rational.

As RHS is rational LHS is also rational i.e

5

is rational.

But this contradicts the fact that

5

is irrational.

This contradiction arose because of our false assumption.

so,

3 + 5 irrational.