Question

Prove that √3+√5 is an irrational
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Answer 1

Step-by-step explanation:

To prove : √3+√5 is irrational.

Let us assume it to be a rational number.

Rational numbers are the ones that can be expressed in q/p form where p,q are integers and q isn't equal to zero.

√3 + √5 = p/q

√3 = p/q -√5

squaring on both sides,

$= > 3 = \frac{ {p}^{2} }{ {q}^{2} } - 2. \sqrt{5} ( \frac{p}{q} ) + 5$

$= > \frac{(2 \sqrt{5p} )}{q} = \frac{2 {q}^{2} - {p}^{2} }{q}$

$= > \sqrt{5} = \frac{2 {q}^{2} - {p}^{2} }{ {q}^{2} } . \frac{q}{2p}$

$= > \sqrt{5} = \frac{(2 {q}^{2} - {p}^{2} )}{2pq}$

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 is irrational.