Question

Which of the following is not an irrational number?
 $
  A.\,\,5 - \sqrt 3 \\
  B.\,\,\sqrt 5 + \sqrt 3 \\
  C.\,\,4 + \sqrt 2 \\
  D.\,\,5 + \sqrt 9 \\
  $

Answer 1

Hint: To find whether given number is a rational or irrational number we will see if there is any simplification possible then we will first do it and then we see that if there is a radical number still present in given number then it will be name irrational and if not then it will be named rational and using this we can get solution of given problem

Complete step-by-step answer:
To find the correct option from given options. We discuss each given option one by one to find the correct option.
In first option we have
 $ 5 - \sqrt 3 $
Here, we see that there is a difference of two numbers $ 5\,\,and\,\,\sqrt 3 $ .
Where number $ 5 $ is a rational number and number $ \sqrt 3 $ is an irrational number.
We know that the sum or difference of a rational and irrational number is always an irrational number.
Therefore, we can say that option (A) is an irrational number and hence not the correct option.

Now, we will discuss option (B)
In this we have $ \sqrt 5 + \sqrt 3 $
Here, it is a sum of two numbers. Which are $ \sqrt 5 $ and $ \sqrt 3 $ .
We see that both terms $ \sqrt 5 $ and $ \sqrt 3 $ are irrational numbers.
We know that the sum or difference of two irrational numbers is also an irrational number.
So, option (B) is an incorrect option.

In option (C) we have $ 4 + \sqrt 2 $ .
Here, we see that the given number is the sum of two numbers.
One number is $ 4 $ and the other is $ \sqrt 2 $ .
Where number $ 4 $ is a rational number and number $ \sqrt 2 $ is irrational number.
We know that the sum or difference of a rational and irrational number is always an irrational number.
Therefore, we can say that option (C) is an irrational number and hence not the correct option.

Now, we will discuss option (D).
In this we have $ 5 + \sqrt 9 $
Which is a sum of two numbers $ 5\,\,and\,\,\sqrt 9 $ .
Here, number $ 5 $ is a rational number and number $ \sqrt 9 $ can also be written as $ \sqrt {{3^2}} or\,\,3 $
Therefore, above written number will become
 $
\Rightarrow 5 + \sqrt 9 = 5 + 3 \\
   = 8 \;
  $
Which is clearly a rational number.
So, we can say that option (D) is not an irrational number.
Hence, the correct option is (D).

Note: Irrational numbers are those numbers which can’t be written in the form of $ \dfrac{p}{q} $ . So, to distinguish between rational and irrational numbers we must first see that is there any simplification possible in given terms. If so then before to say rational or irrational one must do simplification and after it if there is any radical term present in number then we can say that number is irrational otherwise rational.