Question

3 m
216. Prove that 3+25 is irrational. Given that 5 is irrational number.​

Answer 1

Question :–

Prove that 3 + 2√5 is an Irrational.

To Prove :–

• 3 + 2√5 is an Irrational.

Proof :–

Let, 3 + 2√5 = $\sf \frac{a}{b}$ is a rational number.

Where a & b are jntegers, q ≠ 0

→ $2 \sqrt{5} = \frac{a}{b} - 3$

→ $2 \sqrt{5} = \frac{a - 3}{b}$

→ $\sqrt{5} = \frac{a - 3}{2b}$

Since, a & b are integers.

So, (a – 3b) and 2b are also integers. Therefore, $\sf \frac{a - 3b}{2b}$ is a rational number.

But we know that √5 is a Irrational & also given in the question above.

It's L.H.S. = R.H.S.

and R.H.S. = Rational

It is not possible.

Hence, 3 + 2√5 is an Irrational.