Write whether on simplification gives a rational or an irrational number

Question

Write whether on simplification \(\frac{2\sqrt{45}+3\sqrt{20}}{2\sqrt5}\) gives a rational or an irrational number

Answer 1

Given:

number \(\frac{2\sqrt{45}+3\sqrt{20}}{2\sqrt5}\)

To find:

Whether the given number is rational or irrational

Solution:

Factorize 45 and 20.

⇒ \(\frac{2\sqrt{45}+3\sqrt{20}}{2\sqrt5}\) = \(\frac{2\sqrt{3\times 3\times 5}+3\sqrt{2\times 2\times 5}}{2\sqrt5}\)

⇒ \(\frac{2\sqrt{3^2\times 5}+3\sqrt{2^2\times 5}}{2\sqrt5}\)

⇒ \(\frac{2\times 3\sqrt{5}+3\times 2\sqrt{5}}{2\sqrt5}\)

⇒ \(\frac{2\sqrt{5}(3+3)}{2\sqrt5}\)

⇒ 3 + 3 = 6

We know that a rational number is defined as the number which can be written in the form of p/q.

As 6 can be written as 6/1.

So 6 is a rational number.

The given number after simplification gives a rational number.