Question

Between which two integers is the value of radical or the square root of 61 lies?

Answer 1

Hint: Very first we have to find the radical or square root of 61. Now if the given number is a perfect square then we can easily find the root or radical. But since it is not a perfect square, we will use a formula to find the root. Then we will check in which two integers it exactly lies.
Formula used:
Root by approximation is given by,
\[\sqrt {Nearest{\text{ }}perfect{\text{ }}square} + \dfrac{{\left( {Given{\text{ }}number - Nearest{\text{ }}perfect{\text{ }}square} \right)}}{{2\sqrt {Nearest{\text{ }}perfect{\text{ }}square} }}\]

Complete step-by-step answer:
Given is the number 61.
Now the nearest perfect square to 61 is 49 that is square of 7 and 64 that is square of 8.
We will take 49 in our calculations.
Using the formula above,
Root by approximation is given by,
\[\sqrt {Nearest{\text{ }}perfect{\text{ }}square} + \dfrac{{\left( {Given{\text{ }}number - Nearest{\text{ }}perfect{\text{ }}square} \right)}}{{2\sqrt {Nearest{\text{ }}perfect{\text{ }}square} }}\]
\[ = \sqrt {49} + \dfrac{{\left( {61 - 49} \right)}}{{2\sqrt {49} }}\]
Now taking the square root of the number,
\[ = 7 + \dfrac{{12}}{{2 \times 7}}\]
On calculating the fraction term,
\[ = 7 + \dfrac{6}{7}\]
Now dividing by 7 we get,
\[ = 7 + 0.857\]
On adding we get,
\[ = 7.857\]
Thus this is the answer for the number given above.
That is \[\sqrt {61} = 7.857\]
The solution is not finished yet. We have to write between which two integers this radical lies.
So we can observe that the radical lies between 7 and 8.
So, the correct answer is “7.857”.

Note: We can choose the answer in a short time also just from the number itself. But this is only if the options given are with fixed numbers if they j=have a very slight difference then we must use this method. Slight change by I mean that the square of 7 is 49 and that of 8 is 64. Our given number lies in between 49 and 64. So we can say radical lies between 7 and 8.
Also note that we have taken the nearest perfect square number smaller than the given number. But that is not compulsory. We can go for a perfect square number greater than the given number. In that case only the numerator of the fraction will be negative and we need to subtract the fraction value from the perfect square root value.