Answer

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**Hint:**Here we will use the long division method to find the square root.

Long division method is used to divide a large number (usually three digits or more) by a number having two or more digits.

**Complete step-by-step answer:**

Since we have to find the root of 245 correct to two decimal places

Therefore we will write 245 in decimals as:-

\[ \Rightarrow 245.0000\]

Now we will apply a long division method to find its square root.

Applying long division method we get:-

First we will place a bar over the pair of numbers starting from the unit place and also the decimals.

\[ \Rightarrow \overline 2 \overline {45} .\overline {00} \overline {00} \]

Now we will take the largest number as the divisor whose square is less than or equal to 2 then divide and write the quotient i.e. 1.

\[

1\mathop{\left){\vphantom{1{\overline 2 \overline {45} .\overline {00} \overline {00} }}}\right.

\!\!\!\!\overline{\,\,\,\vphantom 1{{\overline 2 \overline {45} .\overline {00} \overline {00} }}}}

\limits^{\displaystyle \,\,\, 1} \\

- 1 \\

\cdots \cdots \cdots \cdots \\

{\text{ }}1 \\

\cdots \cdots \cdots \cdots \\

\]

Now we will bring down 45, which is under the bar, to the right side of the remainder and double the value of the quotient and write it with blank space on the right side. Next, we have to select the largest digit for the unit place of the divisor such that the new number, when multiplied by the new digit at unit’s place, is equal to or less than 145 and put a decimal point in the quotient as there is a decimal in the dividend.

\[

25\mathop{\left){\vphantom{1{145}}}\right.

\!\!\!\!\overline{\,\,\,\vphantom 1{{145}}}}

\limits^{\displaystyle \,\,\, {5.}} \\

{\text{ }} - 125 \\

\cdots \cdots \cdots \cdots \cdots \\

{\text{ 20}} \\

\cdots \cdots \cdots \cdots \\

\]

Again we will bring down 00, which is under the bar, to the right side of the remainder and double the value of the quotient and enter it with blank space on the right side. Next, we have to select the largest digit for the unit place of the divisor such that the new number, when multiplied by the new digit at the unit's place, is equal to or less than 2000.

\[

306\mathop{\left){\vphantom{1{2000}}}\right.

\!\!\!\!\overline{\,\,\,\vphantom 1{{2000}}}}

\limits^{\displaystyle \,\,\, 6} \\

{\text{ }} - 1836 \\

\cdots \cdots \cdots \cdots \cdots \\

{\text{ 164}} \\

\cdots \cdots \cdots \cdots \cdots \\

\]

Again bring down 00, which is under the bar, to the right side of the remainder and double the value of the quotient and enter it with blank space on the right side. Next, we have to select the largest digit for the unit place of the divisor such that the new number, when multiplied by the new digit at unit’s place, is equal to or less than 16400.

\[

3125\mathop{\left){\vphantom{1{16400}}}\right.

\!\!\!\!\overline{\,\,\,\vphantom 1{{16400}}}}

\limits^{\displaystyle \,\,\, 5} \\

{\text{ }} - 15625 \\

\cdots \cdots \cdots \cdots \cdots \\

{\text{ 775}} \\

\cdots \cdots \cdots \cdots \cdots \\

\]

Hence the final quotient is 15.65 correct to two decimal places

Therefore the square root of 245 correct to three decimal places is 15.65

**Note:**The student may make mistakes while selecting the right quotient, so one should follow the steps of the long division method carefully and should continue the process until the remainder comes out to be zero.

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