Question

1. Prove that root 5 - root 3 is not a rational number.

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Answer 1

Step-by-step explanation:

Let √5 - √3 is rational.

Hence, √5 - √3 can be written in the form of (a/b) {a,b are co-primes}

⇒ √5 - √3 = (a/b)

⇒ √5 = (a/b) + √3

On squaring both sides, we get

⇒ 5 = (a²/b²) + (3) + (2a√3/b)

⇒ 2 = (a²/b²) + 2√3(a/b)

⇒ -2√3(a/b) = (a²/b²) - 2

⇒ -2√3(a/b) = (a² - 2b²)/b²

⇒ -2√3(a) = (a² - 2b²)/b

⇒ -√3 = (a² - 2b²)/2ab

⇒ √3 = 2b² - a²/2ab

Here, 2b² - a²/2ab is rational number.

But √3 is irrational number.

Since, Rational ≠ Irrational.

This is a contradiction.

∴ Our assumption is incorrect.

Hence √5 - √3 is not a rational number.

Hope it helps!

Answer 2

Step-by-step explanation:

Let √5 - √3 be rational

√5  - √3 = p/q

√5 = p/q + 3

Squaring on both sides

(√5)^2 = (p/q + √3)^2

5 = (p^2/q^2) + (2√3p/q) + 3

2 = (p^2/q^2) + 2√3p/q

-p^2/q^2 + 2 = √3(p/q)

√3 = (p/q)[2 - p²/b²]

Here, √3 is irrational but (p/q)[2 - p²/b²] is rational.

This is a contradiction.

Therefore, √5 - √3 is irrational.

Hope it helps you.