Prove that √3 + √5 is Irrational

Answer: To prove that √3 +√5 is an irrational number.

Assume that the total of √3 +√ 5 is a rational number.

Now, it can be written in the form a/b:

a/b = √3 + √5 

a and b are coprime numbers

and b ≠ 0

Solving

√3 + √5 = a/b

On squaring both sides, we get,

(√3 + √5)² = (a/b)²

(√3)² + (√5)²+ 2(√5)(√3) = a²/b²

3 + 5 + 2√15 = a²/b²

8 + 2√15 = a²/b²

2√15 = a²/b² – 8

√15 = (a²– 8b²)/2b

Here a and b are integers, then (a²-8b²)/2b is a rational number.

Then √15 is also a rational number.

However, this is incompatible because 15 is an irrational number.

Our assumption is incorrect.

√3 + √5 is an irrational number.

Hence, proved.