Question

Prove that root 5 -2root 3 is anirrational

Answer 1

Hello,

PROOF :-

Let root 5 - 2 root 3 be a rational number

So
It can be expressed in the form of p/q where q≠0 and p, q are Co primes as well as integers.
So,
For some integer

$\sqrt{5} - 2 \sqrt{3} = \frac{p}{q} \\ \\ squaring \: both \: sides \\ {( \sqrt{5} - 2 \sqrt{3} )}^{2} = {( \frac{p}{q} )}^{2} \\ \\ = 5 + 12 - 4 \sqrt{15} = \frac{ {p}^{2} }{ {q}^{2} } \\ = 4 \sqrt{15} = 17 - \frac{ {p}^{2} }{ {q}^{2} } = \frac{17 {q}^{2} - {p}^{2} }{ {q}^{2} } \\ \\ = \sqrt{15} = \frac{17 {q}^{2} - {p}^{2} }{4 {q}^{2} }$
The RHS is a rational number
While root15 (LHS) is not a rational number.

Hence,
It is a contradiction
Our assumption was wrong.

So
Root 5 - 2 root 3 is not a rational number. It is an irrational number.

Hence, proved



Hope this will be helping you ✌️