Question

Write three rational numbers between $\sqrt 3 $ and $\sqrt 5 $ .
A. $1.8,2,2.2$
B. $\sqrt 9 ,5,9$
C. $2,7,1.9$
D. $\sqrt {16} ,5,1$

Answer 1

Hint: Square root of a number is the value which on multiplication by itself gives the original number. Perfect square root does not exist for numbers ending with $2,3,7,8$ in the unit's place. We will first find the values of $\sqrt 3 $ and $\sqrt 5 $, then we will check all the options whether they satisfy the condition or not.

Complete step-by-step solution:
The value of $\sqrt 3 = 1.732$ and $\sqrt 5 = 2.236$ .
The three rational numbers should lie between $\sqrt 3 $ and $\sqrt 5 $.
Here we go with checking all the options as we have to find a correct option, so we check if the given values lie b/w 1.732 and 2.236 or not.
For option A. $1.8,2,2.2$ ,
$1.732 < 1.8 < 2 < 2.2 < 2.236$
Therefore, all the numbers lie between $\sqrt 3 $ and $\sqrt 5 $.
For option B. $\sqrt 9 ,5,9$, $\sqrt 9 = 3, - 3$
$ - 3 < 1.732 < 3 < 2.236 < 5 < 9$
Not all numbers lie between $\sqrt 3 $ and $\sqrt 5 $.
For option C. $2,7,1.9$,
$1.732 < 1.9 < 2 < 2.236 < 7$
Not all numbers lie between $\sqrt 3 $ and $\sqrt 5 $.
For option D. $\sqrt {16} ,5,1$, $\sqrt {16} = 4, - 4$
$ - 4 < 1 < 1.732 < 2.236 < 4 < 5$
Not all numbers lie between $\sqrt 3 $ and $\sqrt 5 $.
Hence, the correct option is A. $1.8,2,2.2$

Note: There are infinitely many numbers between two rational numbers. Where there is a condition given, we must check whether all the numbers lie in between them or not. While taking square root in such cases we must consider both positive and negative numbers.