Question

Prove that the following are irrational.
$\sqrt{3}+\sqrt{5}$

Answer 1

Let √3 + √5 be a rational number , say r

then √3 + √5 = r

On squaring both sides,

(√3 + √5)² = r²

3 + 2 √15 + 5 = r²

8 + 2 √15 = r²

2 √15 = r² - 8

√15 = (r² - 8) / 2

Now (r² - 8) / 2 is a rational number and √15 is an irrational number .

Since a rational number cannot be equal to an irrational number . Our assumption that √3 + √5 is rational wrong .