Question

show that
$( \sqrt[]{3} + 5) {}^{2} \: is \: irrational \: using \: division \: lemma$

Answer 1

let us assume that(√3+2)²is rational
let (√3+2)²=a/b
(√3+2)² =a/b
[a+b]²=a²+2ab+b²
(√3)²+2(√3)(2)+(2)²=[a/b]²
square and root should be cancelled.....
we get.......
3+2√3×2+4=a²/b²
7+2√3×2=a²/b²
7+4√3=a²/b²
4+√3=a²/b²-7
4+√3=a²-7b²/b2
√3=a²-7b²/4b².................eq-1
therefore LHS≠RHS
therefore a, b are integers, LHS of eq-1 √3 is irrational and RHS of eq-1 a²-7b²/4b² is rational..
these contradicts the fact that √3 is irrational
Hence (√3+2)² is an irrational....






hope this helps you & dont forgot to mark me as the brainliest