How to Calculate the Value of the square root of 40?
The following method is used to calculate the square root of 40.
- Square root of 40 by Approximation method
- Square Root of 40 by Long Division Method
- Square Root of 40 by Newton Raphson method
Square Root of 40 by Approximation method
Finding two perfect squares between which 40 falls is the first step.
As we know that \(36\left ( 6^{2} \right )\) and \(49\left ( 7^{2} \right )\) are the two ideal squares that 40 falls between.
40 will therefore have a square root that is greater than 6 but less than 7.
\(6 < \sqrt{40} < 7\)
Consequently, 6 will be the whole number part.
We will use the following formula for the decimal portion:
\(\frac{Given \,no. – Lower\, perfect \, square}{Bigger \, perfect\, square\, – lower\, perfect\, square}\)
\(= \frac{\left ( 40-36 \right )}{49-36} \)
\(= \frac{4}{13} = 0.31\)
Consequently, 6.31 will be approximately equivalent to the square root of 40.
Square Root of 40 by Long Division Method
The long division method can be used to find the square root of 40 by performing the steps listed below.
As seen in the diagram below, write 40. Place a bar on top of the digits starting at position one to begin pairing them in pairs of two.
Find a number that, when multiplied by itself, obtains a result that is less than or equal to 40.
\(6\times 6 = 36\)
Add the divisor that was determined in the previous step to itself and then subtract it from the dividend as we would in a normal division. The remainder will be 4 and the divisor will change to 12.
We at once place a decimal point after the dividend and quotient because there are no more numbers in the dividend. Bring the first pair of zeros down and add three pairs of zeros after the decimal in the dividend part.
Find a number at the unit’s place of divisor that gives a result that is less than 400. Here, 3 will serve as the number.
\(123\times 3 = 369\)
Repeat the process with the following pair of zeros and the final pair of zeros.
Therefore, using the long division method, we can calculate the square root of \(\sqrt{40} = 6.324\)
By Newton Raphson method
\(x = \sqrt{40}\Rightarrow x^{2} =40 \Rightarrow x^{2} -40 =0\)
\(f\left ( x \right ) = x^{2}- 40\)
\(f’\left ( x \right ) =2 x\)
let estimate the initial value for the function
\(x_{0} = 6\)
Now using the formula of root finding by the Newton Raphson method
\(x_{n+1} = x_{n} – \frac{f\left ( x_{n} \right )}{f’\left ( x_{n} \right )}\)
Where,
\(f\left ( x \right ) = x^{2}-40\)
\(f’\left ( x \right ) = 2x\)
Case :1
\(x_{0} =6\)
\(x_{1} =6 – \frac{\left ( -4 \right )}{12} = \frac{98-9}{14}\)
\(x_{1} = 6 +\frac{1}{3} = \frac{18+1}{3}\)
\(x_{1} = \frac{19}{3} = 6.333\)
Case 2
\(x_{2} = 6.333-\frac{0.106889}{12.666}\)
\(x_{2} = \frac{80.1068}{12.666}\)
\(x_{2} = 6.324\)
As a result we can see the value of the square root of 40 is approx. 6.32
