Question

prove
$\sqrt{3} + \sqrt{5}$
is an irrational no.

Answer 1

⚜️⚜️HEY MATE! HERE'S YOUR ANSWER!⚜️⚜️

let √3 + √5 be a rational number = x.

then,

x = √3 + √5

x² = ( √3 + √5 )²

x² = (√3)² + (√5)² + 2(√3)(√5)

x² = 3 + 5 + 2√15

x² = 8 + 2√15

x² - 8 = 2√15

$\frac{ {x}^{2} - 8 }{2}$ = √15 ......(i)

Now,

x is rational,

so, x² is also rational.

=> $\frac{ {x}^{2} - 8 }{2}$ is rational.

=> √15 is rational !

BUT, √15 is irrational.

Thus, we arrive at a contradiction. So, our supposition that √3 + √5 is rational, is WRONG!

Hence, √3 + √5 is an irrational number.

❇️❇️ hope it is helpful ✌️! ❇️❇️