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Question: Write whether \(\dfrac{{2\sqrt {45} + 3\sqrt {20} }}{{2\sqrt 5 }}\) on simplification gives as ratio...

Write whether 245+32025\dfrac{{2\sqrt {45} + 3\sqrt {20} }}{{2\sqrt 5 }} on simplification gives as rational or an irrational.

Explanation

Solution

Here we need to factorize the root numbers which were in the given mathematical expression.To find a factorization of the root number we use LCM method.
On some simplification, we get the solution and find whether it is rational or irrational.
According to the definition of rational number, A number is said to be rational number if it is in the form of  pq, q≠0\;\dfrac{p}{q},{\text{ q}} \ne 0 and p, q∈Ip,{\text{ q}} \in I where I is the set of integers)

Complete step-by-step answer:
It is given that the expression, 245+32025\dfrac{{2\sqrt {45} + 3\sqrt {20} }}{{2\sqrt 5 }}
Take an expression as equation (1)
Now we can write, 245+32025.....→(1)\dfrac{{2\sqrt {45} + 3\sqrt {20} }}{{2\sqrt 5 }}..... \to (1)
Then, we find factors of the root value 45,2045,20 by using LCM (Least Common Multiple) Method.
On finding factors,
45=3×3×545 = 3 \times 3 \times 5
Similarly, we can find factors for
  20=2×2×5\;20 = 2 \times 2 \times 5
Substituting the value of 4545 in the root can be expressed as
We are getting,
∴45=3×3×5=35\therefore \sqrt {45} = \sqrt {3 \times 3 \times 5} = 3\sqrt 5
Similarly, we do this process for 2020
Substituting the value of 2020 in the root can be expressed as
We are getting,
∴20=2×2×5=25\therefore \sqrt {20} = \sqrt {2 \times 2 \times 5} = 2\sqrt 5
Substituting the values of 45\sqrt {45} and 20\sqrt {20} in the equation (1)(1)
We will get,2×35+3×2525\dfrac{{2 \times 3\sqrt 5 + 3 \times 2\sqrt 5 }}{{2\sqrt 5 }}
Now take common factor 252\sqrt 5 in the numerator part, we get
=25(3+3)25= \dfrac{{2\sqrt 5 \left( {3 + 3} \right)}}{{2\sqrt 5 }}
On some simplification we will get the answer
=6, which is the rational number.
Hence, the given expression is a rational number on simplification.

Note:
Definition of irrational number:A number is said to be rational number if it is in the form of   pq, q≠0\;\dfrac{p}{q},{\text{ q}} \ne 0 and p, q∈Ip,{\text{ q}} \in I where I is the set of integers).In this question, it is in factor formation, so we can say this expression is not irrational on simplification.