Question

Which of the following is not a rational number?
A) $\sqrt 2 $
B) $\sqrt 4 $
C) $\sqrt 9 $
D) $\sqrt {16} $

Answer 1

Hint: According to given in the question we have to determine that which of the given number is not rational number but first of all we have to understand about the rational numbers as mentioned below:
Rational numbers: A number that can be made by dividing two integers/numbers (an integer is a number with no fractional part) or we can say that a rational number is a number which can be represented in the form of $\dfrac{p}{q}$where as we all know that p is the nominator number/integer and q is the denominator integer.

Complete step-by-step solution:
Step 1: First of all we have to find the square root of option (A) which is $\sqrt 2 $as mentioned in the solution hint. Hence,
$ \Rightarrow \sqrt 2 = 1.414$
As we can see, the square root of $\sqrt 2 $ is 1.414 which is the decimal representation of $\sqrt 2 $ is non-terminating and non-repeating. Hence we can say that $\sqrt 2 $ is not a rational number.
Step 2: Now, we have to check that $\sqrt 4 $ is the same as the solution step 1 but before that we have to determine the square root as mentioned in the solution hint. Hence,
$
   \Rightarrow \sqrt 4 = \sqrt {2 \times 2} \\
   \Rightarrow \sqrt 4 = 2
 $
As we can see that square root of $\sqrt 4 $is 2 which is clearly a rational number. Hence we can say that $\sqrt 2 $ is a rational number.
Step 3: Now, we have to check that $\sqrt 9 $same as the solution step 1 but before that we have to determine the square root as mentioned in the solution hint. Hence,
$
   \Rightarrow \sqrt 9 = \sqrt {3 \times 3} \\
   \Rightarrow \sqrt 9 = 3
 $
As we can see that square root of $\sqrt 9 $ is 3 which is clearly a rational number. Hence we can say that $\sqrt 9 $ is a rational number.
Step 4: Now, we have to check that $\sqrt {16} $ is the same as the solution step 1 but before that we have to determine the square root as mentioned in the solution hint. Hence,
$
   \Rightarrow \sqrt {16} = \sqrt {4 \times 4} \\
   \Rightarrow \sqrt {16} = 4
 $
As we can see that square root of $\sqrt {16} $ is 4 which is clearly a rational number. Hence we can say that $\sqrt {16} $ is a rational number.
Hence, we can say that $\sqrt 2 $ is not a rational number.

Therefore option (A) is correct.

Note: Number that can be made by dividing two integers/numbers (an integer is a number with no fractional part)
We can determine the square root of a number $\sqrt {{a^2}} $ as we know that $\sqrt a \times \sqrt a = a$ same as the square root of other numbers can be determined.