Question

Estimate the value of square root:\[\sqrt{22}\]

Answer 1

Hint: The square root of \[\sqrt{22}\] can be found by the long division method. In this method, take the largest number as a divisor whose square is less than or equal to the number on the extreme left of the number. The digit on extreme left is the dividend. Divide and write the question.

Formula used:
Formula used here is the division in the long division method.
\[diviso{{r}^{2}}\le dividen{{d}^{2}}\]
\[2\times quotient=newdivisor\]

Complete step by step answer:
In order to find the square root of \[\sqrt{22}\] by a long division method, write the pair of digits from one place. For \[\sqrt{22}\], it is \[22\]. Find a divisor such that its square is less than or equal to \[22\].
\[4{{\left| \!{\overline {\,
 \begin{align}
  & 22.0000 \\
 & 16\downarrow \downarrow \\
 & =600 \\
\end{align} \,}} \right. }^{4}}\]
Then Find the remainder, by subtracting the terms. In this case, we get a remainder \[6\] . Now double the quotient \[4\times 2\] and write in a new divisor’s place .There are no numbers to divide , so we place zeros after a decimal point , so we can do further division. Now bring the zeros down. Find a divisor such that its square is less than or equal to \[600\].
\[86{{\left| \!{\overline {\,
 \begin{align}
  & 60000 \\
 & 516\downarrow \downarrow \\
 & =8400 \\
\end{align} \,}} \right. }^{6}}\]
Writing the next quotient: \[6\]
This gives a new divisor \[86\] such that \[86\times 6=516\]. After doing the division, we get the remainder as\[84\]. There are no numbers to divide, so we place zeros after a decimal point, so we can do further division. Now bring the zeros down. Double the quotient \[2\times 46=92\] and write it in place of the next divisor. Find a divisor such that its square is less than or equal to \[8400\].
\[929{{\left| \!{\overline {\,
 \begin{align}
  & 8400 \\
 & 8361 \\
 & =39 \\
\end{align} \,}} \right. }^{9}}\]
We can stop here as we have found upto two decimal places.
\[\Rightarrow \sqrt{22}=4.69\]

Note: We can find the actual value of $\sqrt{22}$= 4.69041575982343 by the long division method.
Also, The square root of 22 is already in its simplest radical form and cannot be simplified.
Choose the very first divisor very carefully that affects the whole solution.