Question

Which of the following numbers have rational square roots?
(This question has multiple correct options)
A. 2
B. 4
C. 9
D. 11

Answer 1

Hint: We use the concept of square root of a number and calculate square root of each of the given numbers. Since a rational number can be of the form \[\dfrac{p}{1}\] where p is an integer, we find the numbers who satisfy this equation.
* Rational numbers are the numbers that can be written in the form \[\dfrac{p}{q}\] where \[q\]is non-zero and both the numerator and denominator are integers, i.e. the decimal representation of a rational number is either terminating or recurring. Examples: \[\dfrac{1}{2},\dfrac{0}{5},\sqrt 4 = \pm 2,7\]etc.

Complete step-by-step solution:
We solve each part separately and check if the number has a rational square root. We calculate the square root of each number and check if it is rational or not.
A. 2
Let us take square root of the value 2
\[\sqrt 2 \]
Now we calculate the value of square root of 2 using calculator
\[\sqrt 2 = 1.414213...\]
Since the decimal expansion of the square root of the number is non-terminating and non-recurring, the number is not rational.
B. 4
Let us take square root of the value 4
\[\sqrt 4 \]
Now we calculate the value of square root of 4
We can write \[4 = {2^2}\]
\[ \Rightarrow \sqrt 4 = \sqrt {{2^2}} \]
Cancel square root by square power
\[ \Rightarrow \sqrt 4 = \pm 2\]
Since the number comes out to be an integer which can be expressed in the form of rational number i.e. \[\dfrac{{ \pm 2}}{1}\]
\[ \Rightarrow \]4 has rational square roots.
C. 9
Let us take square root of the value 9
\[\sqrt 9 \]
Now we calculate the value of square root of 9
We can write \[9 = {3^2}\]
\[ \Rightarrow \sqrt 9 = \sqrt {{3^2}} \]
Cancel square root by square power
\[ \Rightarrow \sqrt 9 = \pm 3\]
Since the number comes out to be an integer which can be expressed in the form of rational number i.e. \[\dfrac{{ \pm 3}}{1}\]
\[ \Rightarrow \]9 has rational square roots.
D. 11
Let us take square root of the value 11
\[\sqrt {11} \]
Now we calculate the value of square root of 11 using calculator
\[\sqrt {11} = 3.316624...\]
Since the decimal expansion of the square root of the number is non-terminating and non-recurring, the number is not rational.

\[\therefore \]Option B and C are correct.

Note: Many students get confused with the meaning of the question as they usually think rational numbers only \[\dfrac{p}{q}\] form and don’t know that decimal representation should be such. They think even \[\sqrt x \] can be written as \[\dfrac{{\sqrt x }}{1}\] which is a rational number. This is a wrong concept, we have to check if the decimal representation of square root is terminating or recurring or not.