Question

Prove that √3+√5is an irrational number by contradiction method

Answer 1

Let √3 + √5 be a rational number.
A rational number can be written in the form of p/q where p,q are integers.

√3 + √5 = p/q
√5 = p/q - √3

Squaring on both sides,
√5² = [p/q - √3]²
5 = p²/q² + √3² - 2(p/q)(√3)
5 = p²/q² + 3 - 2√3p/q
p²/q² - 2√3p/q = 5 - 3
p²/q² - 2 = 2√3p/q
(p²-2q²)/q² = 2√3p/q
(p²-2q²)/q² × q/2p = √3
(p²-2q²)/2pq = √3

p,q are integers then (p²-2q²)/2pq must be a rational number.
Then √3 is also a rational number.
But this contradicts the fact that √3 is an irrational number.
Hence,our supposition is false.

So, √3+√5 is an irrational number.
Hence proved!