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**Hint:**Here, we will solve this question by using a long division method to find the square root. We will pair each number except the first digit and divide it with a suitable number. We will find the square root correct to 1 decimal place and then we will write the required final value.

**Complete step by step solution:**

As we need to find the square root of the given number i.e. 137, so we will write 137 as 137.0000, as we take pairs of digits while calculating the square root.

In the long division method of finding the square root, we have to first find the square of a number which is closest to the first digit of a number i.e. 1.

We know that \[{1^2} = 1 = 1\], therefore, we will divide 137 by 1 to obtain

\[\begin{array}{*{20}{r}}1\\\begin{array}{l}1\left| \!{\overline {\,

{137.0000} \,}} \right. \\\underline { - 1} \\0\end{array}\end{array}\]

Thus, the first digit in \[\sqrt {137} \] should be 1.

We have got the remainder as 0 and we will take down the next two digits after the decimal place. Now, the ones place of the divisor is of the form \[1 \times 2 = 2\] and we will try to find out a number of the form \[2x \times x\] is lower than but closest to 37.

As \[21 \times 1 = 21 < 37\] therefore, we will divide 37 by 21 to obtain

\[\begin{array}{*{20}{r}}{}&1\\{21}&\begin{array}{l}\left| \!{\overline {\,

\begin{array}{l}37\\\underline { - 21} \end{array} \,}} \right. \\16\end{array}\end{array}\]

Thus, the second digit in \[\sqrt {137} \] should be 1.

We have got the remainder as 16 and we will take down the next two digits after the decimal place. Now, the tens place of the divisor is of the form \[20 + 1 \times 2 = 22\] and we will try to find out a number of the form \[22x \times x\] is lower than but closest to 1600

As \[227 \times 7 = 1589 < 1600\] therefore, we will divide 1600 by 227 to obtain

\[\begin{array}{*{20}{r}}{}&7\\{227}&{\left| \!{\overline {\,

\begin{array}{l}1600\\\underline { - 1589} \\11\end{array} \,}} \right. }\end{array}\]

Thus, the first digit after the decimal point in \[\sqrt {137} \] should be 7.

**The square root of the number 137 is \[11.7\].**

**Note:**A square root of a number is defined as the number which when multiplied by itself gives the original number. In other words, when we multiply that number twice, we will get the value that is equal to the given number. We can check if we have found the correct answer or not by multiplying the obtained square root twice.

\[\begin{array}{*{20}{r}}{11.7}\\{ \times 11.7}\\\hline{819}\\{117 \times }\\{117 \times \times }\\\hline{136.89}\end{array}\]

After multiplying we get the numbers as \[136.89\] which is approximately 137.

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