Question

Find the square root of each of the following correct to three places of decimal.
  (i) 5
 (ii) 7
  (iii) 17
  (iv) 20
  (v) 66
  (vi) 427
  (vii) \[1.7\]
  (viii) \[23.1\]
  (ix) \[2.5\]
   (x) \[237.615\]

Answer 1

Hint: Here, we need to find the square roots of the given numbers correct to three places of decimal. We will use a long division method of finding square roots to calculate the required square roots. We should remember that each pair of numbers in the dividend will give one digit of the square root. Also, remember that we can add any number of decimal places with digit 0 at the end of a number.


Complete step-by-step answer:
We will use the long division method to find the square roots of the given numbers.
(i)
We have to find the square root of 5 upto three decimal places.
First, we will group the digits in pairs, starting from the units place.
We will rewrite \[5\] as \[5.000000\] because each pair after the decimal gives one decimal place of the square root.
Pairing the numbers, we get \[\underline 5 .\underline {00} \underline {00} \underline {00} \].
Now, we have to find the number, which when multiplied by itself, is less than or equal to the first pair.
The first pair is 5. Therefore, the number is 2.
Writing this in form of long division, we get
\[\begin{array}{l}{\rm{ }}2\\\begin{array}{*{20}{c}}2\\{}\\{}\end{array}\overline {\left| \begin{array}{l}\underline 5 .\underline {00} \underline {00} \underline {00} \\\underline {4{\rm{ }}} \\100\end{array} \right.} \end{array}\]
Now, we will bring down the next pair 00 to get the next dividend is 100.
First, we will add 2 and 2 to get 4.
Then, we need to find a number with 4 at ten’s place such that when multiplied by the number at unit’s place, the number becomes less than or equal to the new dividend.
Thus, we get the number 42, since \[42 \times 2 = 84\].
Therefore, we continue the long division as
\[\begin{array}{l}{\rm{ }}2.2\\\begin{array}{*{20}{c}}2\\{}\\{42}\\{}\\{}\end{array}\overline {\left| \begin{array}{l}\underline 5 .\underline {00} \underline {00} \underline {00} \\\underline {4{\rm{ }}} \\100\\\underline {{\rm{ 84 }}} \\{\rm{ 1600}}\end{array} \right.} \end{array}\]
We added a decimal in the quotient because the dividend 100 had a pair after the decimal.
Again, bringing down the next pair of 00, we get the next dividend as 1600.
First, we will add 42 and 2 to get 44.
Then, we need to find a number with 4 at hundred’s and ten’s place such that when multiplied by the number at unit’s place, the number becomes less than or equal to the new dividend.
Thus, we get the number 443, since \[443 \times 3 = 1329\].
Therefore, we continue the long division as
\[\begin{array}{l}{\rm{ }}2.23\\\begin{array}{*{20}{c}}2\\{}\\{42}\\{}\\{443}\\{}\\{}\end{array}\overline {\left| \begin{array}{l}\underline 5 .\underline {00} \underline {00} \underline {00} \\\underline {4{\rm{ }}} \\100\\\underline {{\rm{ 84 }}} \\{\rm{ 1600}}\\\underline {{\rm{ 1329 }}} \\{\rm{ 27100}}\end{array} \right.} \end{array}\]
Now, we will bring down the final pair of 00 to get the next dividend 27100.
First, we will add 443 and 3 to get 446.
Then, we need to find a number with 4 at thousand’s place, 4 at hundred’s place and 6 at ten’s place such that when multiplied by the number at unit’s place, the number becomes less than or equal to the new dividend.
Thus, we get the number 4466, since \[4466 \times 6 = 26796\].
Therefore, we continue the long division as
\[\begin{array}{l}{\rm{ }}2.236\\\begin{array}{*{20}{c}}2\\{}\\{42}\\{}\\{443}\\{}\\{4466}\\{}\\{}\end{array}\overline {\left| \begin{array}{l}\underline 5 .\underline {00} \underline {00} \underline {00} \\\underline {4{\rm{ }}} \\100\\\underline {{\rm{ 84 }}} \\{\rm{ 1600}}\\\underline {{\rm{ 1329 }}} \\{\rm{ 27100}}\\\underline {{\rm{ 26796}}} \\\underline {{\rm{ 304}}} \end{array} \right.} \end{array}\]
Therefore, we get the square root of 5 up to three decimal places as \[2.236\].
Similarly, we can find the square roots of other numbers.
(ii)
We have to find the square root of 7 upto three decimal places.
Using long division to find the square root of 7, we get
\[\begin{array}{l}{\rm{ }}2.645\\\begin{array}{*{20}{c}}2\\{}\\{46}\\{}\\{524}\\{}\\{5285}\\{}\\{}\end{array}\overline {\left| \begin{array}{l}\underline 7 .\underline {00} \underline {00} \underline {00} \\\underline {4{\rm{ }}} \\300\\\underline {{\rm{276 }}} \\{\rm{ 2400}}\\\underline {{\rm{ 2096 }}} \\{\rm{ 30400}}\\\underline {{\rm{ 26425}}} \\\underline {{\rm{ 3975}}} \end{array} \right.} \end{array}\]
Therefore, we get the square root of 7 up to three decimal places as \[2.645\].
(iii)
We have to find the square root of 17 upto three decimal places.
Using long division to find the square root of 17, we get
\[\begin{array}{l}{\rm{ 4}}.123\\\begin{array}{*{20}{c}}4\\{}\\{81}\\{}\\{822}\\{}\\{8243}\\{}\\{}\end{array}\overline {\left| \begin{array}{l}\underline {17} .\underline {00} \underline {00} \underline {00} \\\underline {{\rm{16 }}} \\{\rm{ 1}}00\\\underline {{\rm{ 81 }}} \\{\rm{ 1900}}\\\underline {{\rm{ 1644 }}} \\{\rm{ 25600}}\\\underline {{\rm{ 24729}}} \\\underline {{\rm{ 871}}} \end{array} \right.} \end{array}\]
Therefore, we get the square root of 17 up to three decimal places as \[{\rm{4}}.123\].
(iv)
We have to find the square root of 20 upto three decimal places.
Using long division to find the square root of 20, we get
\[\begin{array}{l}{\rm{ 4}}.472\\\begin{array}{*{20}{c}}4\\{}\\{84}\\{}\\{887}\\{}\\{8942}\\{}\\{}\end{array}\overline {\left| \begin{array}{l}\underline {20} .\underline {00} \underline {00} \underline {00} \\\underline {{\rm{16 }}} \\{\rm{ 4}}00\\\underline {{\rm{ 336 }}} \\{\rm{ 6400}}\\\underline {{\rm{ 6209 }}} \\{\rm{ 19100}}\\\underline {{\rm{ 17884}}} \\\underline {{\rm{ 1216}}} \end{array} \right.} \end{array}\]
Therefore, we get the square root of 20 up to three decimal places as \[{\rm{4}}.472\].
(v)
We have to find the square root of 66 upto three decimal places.
Using long division to find the square root of 66, we get
\[\begin{array}{l}{\rm{ 8}}.124\\\begin{array}{*{20}{c}}8\\{}\\{161}\\{}\\{1622}\\{}\\{16244}\\{}\\{}\end{array}\overline {\left| \begin{array}{l}\underline {66} .\underline {00} \underline {00} \underline {00} \\\underline {{\rm{64 }}} \\{\rm{ 2}}00\\\underline {{\rm{ 161 }}} \\{\rm{ 3900}}\\\underline {{\rm{ 3244 }}} \\{\rm{ 65600}}\\\underline {{\rm{ 64976}}} \\\underline {{\rm{ 624}}} \end{array} \right.} \end{array}\]
Therefore, we get the square root of 66 up to three decimal places as \[{\rm{8}}.124\].
(vi)
We have to find the square root of 427 upto three decimal places.
Using long division to find the square root of 427, we get
\[\begin{array}{l}{\rm{ }}20.663\\\begin{array}{*{20}{c}}2\\{}\\{40}\\{}\\{406}\\{}\\{4126}\\{}\\{41323}\\{}\\{}\end{array}\overline {\left| \begin{array}{l}\underline 4 \underline {27} .\underline {00} \underline {00} \underline {00} \\\underline {4{\rm{ }}} \\{\rm{ }}27\\\underline {{\rm{ 0 }}} \\{\rm{ 2700}}\\\underline {{\rm{ 2436 }}} \\{\rm{ 26400}}\\\underline {{\rm{ 24756 }}} \\{\rm{ 164400}}\\\underline {{\rm{ 123969}}} \\\underline {{\rm{ 40431}}} \end{array} \right.} \end{array}\]
Therefore, we get the square root of 427 up to three decimal places as \[20.663\].
(vii)
We have to find the square root of \[1.7\] upto three decimal places.
Using long division to find the square root of \[1.7\], we get
\[\begin{array}{l}{\rm{ 1}}.303\\\begin{array}{*{20}{c}}1\\{}\\{23}\\{}\\{260}\\{}\\{2603}\\{}\\{}\end{array}\overline {\left| \begin{array}{l}\underline 1 .\underline {70} \underline {00} \underline {00} \\\underline {{\rm{1 }}} \\{\rm{ }}70\\\underline {{\rm{ 69 }}} \\{\rm{ 100}}\\\underline {{\rm{ 0 }}} \\{\rm{ 10000}}\\\underline {{\rm{ 7809}}} \\\underline {{\rm{ 2191}}} \end{array} \right.} \end{array}\]
Therefore, we get the square root of \[1.7\] up to three decimal places as \[{\rm{1}}.303\].
(viii)
We have to find the square root of \[23.1\] upto three decimal places.
Using long division to find the square root of \[23.1\], we get
\[\begin{array}{l}{\rm{ 4}}.806\\\begin{array}{*{20}{c}}4\\{}\\{88}\\{}\\{960}\\{}\\{9606}\\{}\\{}\end{array}\overline {\left| \begin{array}{l}\underline {23} .\underline {10} \underline {00} \underline {00} \\\underline {{\rm{16 }}} \\{\rm{ 71}}0\\\underline {{\rm{ 704 }}} \\{\rm{ 600}}\\\underline {{\rm{ 0 }}} \\{\rm{ 60000}}\\\underline {{\rm{ 57636}}} \\\underline {{\rm{ 2364}}} \end{array} \right.} \end{array}\]
Therefore, we get the square root of \[23.1\] up to three decimal places as \[{\rm{4}}.806\].
(ix)
We have to find the square root of \[2.5\] upto three decimal places.
Using long division to find the square root of \[2.5\], we get
\[\begin{array}{l}{\rm{ 1}}.581\\\begin{array}{*{20}{c}}1\\{}\\{25}\\{}\\{308}\\{}\\{3161}\\{}\\{}\end{array}\overline {\left| \begin{array}{l}\underline 2 .\underline {50} \underline {00} \underline {00} \\\underline {{\rm{1 }}} \\{\rm{1}}50\\\underline {{\rm{125 }}} \\{\rm{ 2500}}\\\underline {{\rm{ 2464 }}} \\{\rm{ 3600}}\\\underline {{\rm{ 3161}}} \\\underline {{\rm{ 439}}} \end{array} \right.} \end{array}\]
Therefore, we get the square root of \[2.5\] up to three decimal places as \[{\rm{1}}.581\].
(x)
We have to find the square root of \[237.615\] upto three decimal places.
Using long division to find the square root of \[237.615\], we get
\[\begin{array}{l}{\rm{ 15}}.414\\\begin{array}{*{20}{c}}1\\{}\\{25}\\{}\\{304}\\{}\\{3081}\\{}\\{30824}\\{}\\{}\end{array}\overline {\left| \begin{array}{l}\underline 2 \underline {37} .\underline {61} \underline {50} \underline {00} \\\underline {{\rm{1 }}} \\137\\\underline {{\rm{125 }}} \\{\rm{ 1261}}\\\underline {{\rm{ 1216 }}} \\{\rm{ 4500}}\\\underline {{\rm{ 3081 }}} \\{\rm{ 141900}}\\\underline {{\rm{ 123296}}} \\\underline {{\rm{ 18604}}} \end{array} \right.} \end{array}\]
Therefore, we get the square root of \[237.615\] up to three decimal places as \[{\rm{15}}.414\].

Note: We should follow the steps properly so as to avoid the common mistakes.
For example: In part (i), a common mistake is to not add 42 and 2, to get 44 in the next divisor.
If 42 and 2 is not added, then the next divisor would be a number with 4 at hundred’s, 2 at ten’s place.
Thus, we will get the number 423, since \[423 \times 3 = 1269\].
The final long division would look like
\[\begin{array}{l}{\rm{ }}2.237\\\begin{array}{*{20}{c}}2\\{}\\{42}\\{}\\{423}\\{}\\{4267}\\{}\\{}\end{array}\overline {\left| \begin{array}{l}\underline 5 .\underline {00} \underline {00} \underline {00} \\\underline {4{\rm{ }}} \\100\\\underline {{\rm{ 84 }}} \\{\rm{ 1600}}\\\underline {{\rm{ 1269 }}} \\{\rm{ 33100}}\\\underline {{\rm{ 29869}}} \\\underline {{\rm{ 3231}}} \end{array} \right.} \end{array}\]
Here, you can observe that the square root is \[2.237\] instead of \[2.236\] which is incorrect.