Question

Find the value of $\sqrt {11} $ correct up to three places of decimals?

Answer 1

Hint: Given the numbers in the form of a radical expression. First, we will rewrite the number inside the radical by placing the decimal after the number and add six zeros. Then, we will pair the numbers. Then, use the long division method to find the square root of the number in which first we have to find the number which is multiplied by itself gives less than or equal value to the first pair. Continue the process until we get three digits after the decimal.

Complete step by step answer:
First, we will rewrite the given expression.
$ \Rightarrow 11.000000$
Now, pairing the numbers , we get:
\[ \Rightarrow \underline {11} .\underline {00} {\text{ }}\underline {00} {\text{ }}\underline {00} \]
Now, the first pair is \[11\]. Find the number which is multiplied by itself and gives less than or equal value to \[11\]. Subtract the number from the first pair and bring down the next pair.
\[3)\overline {11.000000} (3.\]
    \[\underline {9{\text{ }}} \]
      \[200\]
Thus, the digit before the decimal place is \[3\]
Now, add \[3 + 3 = 6\] which is the new divisor and find the number of the form \[6x\] such that \[6x \times x\] must be lower than or equal to \[200\]:
\[63)\overline {200} (3\]
    \[\underline {189{\text{ }}} \]
      \[11\]
Thus, the digit at first place after the decimal is \[3\]
Bring down the next pair of \[00\]and add \[63 + 3 = 66\] which is the new divisor and find the number of the form \[66x\] such that \[66x \times x\] must be lower than or equal to \[1100\]:

\[661)\overline {1100} (1\]
    \[\underline {661{\text{ }}} \]
      \[439\]
Thus, the digit at second place after the decimal is \[1\]
Bring down the next pair of \[00\]and add \[331 + 331 = 662\] which is the new divisor and find the number of the form \[662x\] such that \[662x \times x\] must be lower than or equal to \[43900\]:
\[6626)\overline {43900} (6\]
         \[\underline {39756} \]
             \[4144\]
Thus, the digit at third place after the decimal is \[6\]
Thus, the value of \[\sqrt {11} \approx 3.316\]

Hence the value of \[\sqrt {11} \] correct up to three places of decimals is $3.316$

Note: In such types of questions students mainly get confused in applying the formula. As they don't know which formula they have to apply. While applying the long division method students mainly get confused to find out the number which is multiplied by itself and is equal to or less than the first pair of dividends.