Question

How do you find the square root of \[\dfrac{1}{{121}}\]?

Answer 1

Hint: Here, we will find the square root of the numerator and the denominator separately and then divide them to give the square root of the given rational number. A square root of a number is a factor which when multiplied by itself gives the original number.

Complete step by step solution:
We know that the square of a number is obtained by multiplying the number by itself.
For example, when 2 it multiplied by itself, we get 4 i.e., \[2 \times 2 = 4\].
The square root of 4 is 2 i.e., \[\sqrt 4 = 2\].
Now the square root can either be positive or negative. This means that the square root of 4 can either be 2 or \[ - 2\] because \[( - 2) \times ( - 2) = 4\].
Now, we have to find the square root of \[\dfrac{1}{{121}}\]. Let us find the square root of the numerator and denominator separately.
The numerator is 1. We know that either 1 or \[( - 1)\] multiplied by itself gives 1.
So, \[\sqrt 1 = \pm 1\]
Here the symbol \[ \pm \] denotes that the result can be either positive or negative.
We now have to find the square root of \[121\].
We can observe that the unit digit of \[121\] is 1. So, the unit digit of the square root must be either 1 or 9, because a square of 1 is 1 and a square of 9 is 81.
We also know that 121 is greater than 100 which is \[{10^2}\]. So, the square root must be greater than 10. Now, between 10 and 20, there are two numbers whose units digit is either 1 or 9, which are 11 and 19.
Let us find the squares of 11 and 19.
We can see that \[{11^2} = 121\] and \[{19^2} = 361\].
Thus, the required square root of 121 is 11. So,
 \[\sqrt {121} = \pm 11\]

Therefore, the square root of \[\dfrac{1}{{121}}\] is
\[\sqrt {\dfrac{1}{{121}}} = \pm \dfrac{1}{{11}}\]

Here \[\dfrac{1}{{11}}\] is positive if both 1 and 11 are of the same sign and negative if 1 and 11 are of different signs.

Note:
In the given problem, we have found the square root by identifying the unit digit of the given square. If the units digit of a square is 1, then the units digit of the square root is either 1 or 9. Similarly, if the unit digit of a square is 4, then the unit digit of the square root must be either 2 or 8. If the units digit of a square is 6, then the units digit of the square root is either 4 or 6. Further, If the units digit of a square is 9, then the units digit of the square root is either 3 or 7. Also, if the unit digit of a square is either 5 or 0, then the unit digit of the square root is 5 or 0 respectively.