Math, asked by Ritvikskr2084, 1 year ago

Prove that root 5 -2root 3 is anirrational

Answers

Answered by skh2
0
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 ✌️

Similar questions