Question

1.prove that
$\sqrt{3} + \sqrt{4}$
is an irrational number.
​

Answer 1

On the contrary let us assume that √3+√4 is a rational

i.e.,√3+√4=p/q [p,q€N and q≠0]

Squaring on both sides,

(√3+√4)²=p²/q²

→(7+2√12)=p²/q²

$\implies \: \sf{2 \sqrt{12} } = \frac{p {}^{2} }{q {}^{2} } - 7 \\ \\ \sf{ \implies \: \: \sqrt{12} = \frac{p {}^{2} - 7q {}^{2} }{2q {}^{2} } }$

√12 is an irrational whereas the expression on RHS is a rational

Hence,LHS≠RHS

Thus,√3+√4 is an irrational