Question

Prove that √5-√3 is irrational

Answer 1

Hello Mate!

Thanks for the question!

$let \: x = \sqrt{5} - \sqrt{3} \\ {x}^{2} = {( \sqrt{5 } - \sqrt{3} )}^{2} \\ {x}^{2} = 5 + 3 - 2 \times \sqrt{5} \times \sqrt{3} \\ {x}^{2} = 8 - 2 \sqrt{15} \\ {x}^{2} - 8 = - 2 \sqrt{15} \\ - (8 - {x}^{2} ) = - 2 \sqrt{15} \\ 8 - {x}^{2} = 2 \sqrt{15} \\ \frac{8- {x}^{2}}{2} = \sqrt{15}$

Here, ( x^2 - 8 ) / 2 is rational number, and root 15 is rational number. But contradiction say that root 15 is irrational.

Hence proved

Hope it helps☺!✌