Question

What is the square root of \[96.04\] using long division method.

Answer 1

Hint: First of all, let us briefly see how we find square roots using the long division method.
STEP 1 – We need to First Separate the digits from right to left once in two digits.
STEP 2 – Now, if the leftmost digit is a pair or single number, we will have to think of a number such that it is less than or equal to the pair or number at the left most side, when multiplied with itself.
STEP 3 – Now, After obtaining the remainder, we will bring down the next pair and then, we will double the quotient we will have at that time and make it as the starting of the next divisor and then think of a number such that when that number is added to ten times the number we obtained after doubling the quotient and then multiplied by the number we thought should be less than the dividend at that time.
STEP 4 – Now, we will bring down the new pair and then we will have a new quotient and sp, we will repeat the same procedure.

Complete step-by-step solution:
We need to find the square root of \[96.04\]
First of all we must separate the pairs from right to left. i.e. \[\overline {96} .\overline {04} \]
Now, we need to find a number such that when we multiply the number by itself, it should be less than 96.
We see that the number is \[9\] because if we multiply \[10\] by itself then we will get \[100\]which is greater than \[96\]. So, we choose \[9\]as our divisor.
\[9|\overline {96} .\overline {04} |9\]
\[|81\]
\[|\overline {15} \]
Now, bringing down the new pair and then doubling the quotient we have, we get,
 \[9|\overline {96} .\overline {04} |9.\]
 \[|81\]\[ \downarrow \downarrow \]
\[18\] \[|\overline {15} \overline {04} \]
Now, thinking of a number such that it is when added to \[10 \times 18\]and then multiplied with that number gives less than or equal to 1504. Also, we have decimals in between. So, we will add decimal in the quotient and then add the number we found out in the quotient.
We see that that number is \[8\] because, \[(180 + 9) \times 9 = 189 \times 9 = 1701\], which is greater than \[1504\].
Therefore, the new divisor is \[180 + 8 = 188\] and when it is multiplied by \[8\], it gives \[188 \times 8 = 1504\], which is equal to the dividend.
\[9|\overline {96} .\overline {04} |9.8\]
\[|81\]\[ \downarrow \downarrow \]
\[188\] \[|\overline {15} .\overline {04} \]
\[|1504\] (Ignoring the decimal in the dividend)
\[|\overline {0000} \]
Hence, the square root of \[96.04\] is \[9.8\]

Note: We need to be very careful at every step. We have to add decimal to the quotient once the calculation with the integral part is over. Also, we need to keep in mind that after obtaining the new divisor, we have to multiply that divisor with the one digit of the divisor only, not the whole quotient. If we are left with the remainder, we need to consider that as well in the next dividend. Also, we need to be careful that we have to bring down a pair always, no single digit can be brought alone.