Question

how to prove that root 3+root5are rational no.
$how \: to \: prove \: that \: \sqrt \sqrt{ \sqrt{3} } + \sqrt{5are \: rational \: numbers$

Answer 1

Hi Friend ✋✋✋

To prove : 3 + √5 is irrational.

Proof :

let us assume that 3 + √5 is a rational number.

there exists two integers a and b, where a and b are co-primes

such that

$3 + \sqrt{5} = \frac{a}{b} \\ \\ \sqrt{5} = \frac{a}{b} - 3 \\ \\ \sqrt{5} = \frac{a - 3b}{b} \\ \\ \frac{a - 3b}{b} \: is \: an \: rational \: no. \\ \\ it \: means \: \sqrt{5} \: is \: rational \\ but \: we \: know \: that \: \sqrt{5} is \: irrational \\ \\ it \: contradicts \: the \: fact \: that \: \sqrt{5} \: is \: irrational$

so , 3 - √5 is irrational

Answer 2

Kindly refer to the above attachment

Hope it helps you :)

Regards,
Shobana
^_^