Question

Prove that 2+7√3is irrational?

Answer 1

Answer:

TO PROVE : 2+7√5 is an IRRATIONAL number.

PROOF:

$let \: \: 2 + 7 \sqrt{5} \: \: \: is \: a \: rational \: number \: which \: \\ can \: be \: expressed \: in \: the \: frm \: of \: \frac{p}{q} \: where \: \\ p \: and \: q \: are \: co \: prime.$

$2 + 7 \sqrt{5 } \: = \frac{p}{q} \\ \\ 7 \sqrt{5} = \frac{p}{q} - 2 \\ \\ 7 \sqrt{5} = \frac{p - 2q}{q} \\ \\ \sqrt{5} = \frac{p - 2q}{7q} \\ \\ this \: shows \: that \: \sqrt{5} \: is \: a \: rational \: but \: it \: is \: \\ an \: irrational \: number. \\ \\ thus \: our \: contradiction \: was \: wrong.$

Thus 2+√5 is an IRRATIONAL number.

in this way we can prove that it is an irrational number