QUESTION

# Between which two consecutive numbers does $\sqrt {95}$ lie?(a) 7 and 8(b) 8 and 9(c) 9 and 10(d) 10 and 11

Hint: Find the consecutive perfect squares such that 95 lies in between these two perfect squares. Then take the square root of the numbers to find the two consecutive numbers between which the irrational number $\sqrt {95}$ lies.

A square number or a perfect square is an integer made by squaring an integer. For example, 1, 4, 9 all are perfect square numbers.
$\sqrt {95}$ is not a perfect square because it is not a square of any number. $\sqrt {95}$ is also an irrational number.
We need to find the two consecutive integers between which the number $\sqrt {95}$ lies.
For that, first, we find the two consecutive squares between which the number 95 lies.
Hence, the perfect square nearest to 95 and greater than 95 is 100 which is square of 10.
The perfect square nearest to 95 and lesser than 95 is 81 which is square of 9.
Hence, we have the following:
81 < 95 < 100
Taking square root for all three numbers, the inequality is preserved. Hence, we have:
$\sqrt {81} < \sqrt {95} < \sqrt {100}$
The square root of 81 is 9 and the square root of 100 is 10, hence, we have:
$9 < \sqrt {95} < 10$
Hence, $\sqrt {95}$ lies between the consecutive numbers 9 and 10.
Hence, the correct answer is option (c).

Note: You can also find the value of $\sqrt {95}$ by using long division method to find the value of $\sqrt {95}$ up to one decimal place and find the two consecutive numbers between which $\sqrt {95}$ lies.