Question

Find the square root of the following number \[19.5364\]

Answer 1

Hint: A square root of a number is a value that, when multiplied by itself, gives the number
Example: -\[4 \times 4 = 16\], So a square root of \[16\] is 4.
The symbol is \[\sqrt {} \] which always means the positive square root


Complete step by step solution:
How do we find the square root of a number in the decimal form are explained in the following steps:
To find the square root of a decimal number we put bars on the integral part of the number in the usual manner. And place bars on the decimal part on every pair of digits beginning with the first decimal place and proceed as usual.
Taking example of \[\overline {17} .\overline {64} \].
Taking the respective pairs and applying the estimation of square root, we get:
Now we see that the leftmost bar is 17 and 17 lies between \[{4^2} < 17 < {5^2}\]. Take this number as the divisor and the number under the leftmost bar as the dividend i.e. 17 is divided by 4 and we got the remainder.

The remainder is 1. Write the number under the next bar (i.e. \[64\]) to the right of this remainder.
Double the divisor and enter it with a blank on its right since \[64\]is the decimal part so put a decimal point in the quotient as well.

We know \[82 \times 2 = 164\], hence the new digit is \[2\]
On dividing 164 by 8, we get:
Since the remainder is 0 and no bar left therefore, we get:
\[\sqrt {17.64} = 4.2\]

We see that \[(84 \times 4) < 353 < (85 \times 5)\]using \[(84 \times 4)\] as divisor
Remainder is\[17\]. Write the number under the next bar (i.e. \[64\]) to the right of this number to get \[1764\].
Adding 4 to 84 we get 88, we know that \[882 \times 2 = 1764\], there the new digit is\[2\]. Divide and get the remainder.
Since the remainder is 0 and no bar left there\[\sqrt {19.5364} = 4.42\]

Note:-Negative numbers don't have a real square root.Also, the square of a real number is always positive.