Question

Prove that ✓3-✓5 is an irrational number

Answer 1

Answer: Step-by-step explanation:

Let us assume that√3 - √5 is a rational number.

So it can be written in the form a/b

√3 - √5 = a/b

Here a and b are coprime numbers and b ≠ 0

√3 - √5 = a/b

On squaring both sides we get,

(√3 - √5)² = (a/b)²

√3² + √5² - 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

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

Then √15 is also a rational number.

But this contradicts the fact that √15 is an irrational number.

Therefor our assumption is incorrect

√3 - √5 is an irrational number.

Hence, proved.