Using the division method find the square root of :
(i) `605.16`
(ii) `0.000729`

Answer

To find the square roots of the given numbers using the division method, we will follow a systematic approach. Let's solve each part step by step.

### Part (i): Finding the square root of 605.16

1. **Pair the digits**: Start pairing the digits from the decimal point. For 605.16, we pair as (6)(05)(16).

2. **Find the largest square**: The largest perfect square less than or equal to 6 is 4 (which is \(2^2\)). Write 2 above the pair (6).

\[ \begin{array}{c|c} 2 & 4 \\ \hline 6 & 05 \\ \end{array} \]

3. **Subtract and bring down**: Subtract 4 from 6, which gives 2. Bring down the next pair (05) to make it 205.

4. **Double the quotient**: Double the number above (2) to get 4. Now we need to find a digit \(x\) such that \(4x \times x\) is less than or equal to 205.

5. **Trial and error**:
- If \(x = 5\): \(45 \times 5 = 225\) (too high)
- If \(x = 4\): \(44 \times 4 = 176\) (valid)

Write 4 above the pair (05).

\[ \begin{array}{c|c} 2 & 4 \\ \hline 6 & 05 \\ 4 & 176 \\ \end{array} \]

6. **Subtract again**: Subtract 176 from 205, which gives 29. Bring down the next pair (16) to make it 2916.

7. **Double the new quotient**: Double the number above (24) to get 48. Now we need to find a digit \(y\) such that \(48y \times y\) is less than or equal to 2916.

8. **Trial and error**:
- If \(y = 6\): \(486 \times 6 = 2916\) (valid)

Write 6 above the pair (16).

\[ \begin{array}{c|c} 2 & 4 & 6 \\ \hline 6 & 05 & 16 \\ 4 & 176 & 2916 \\ \end{array} \]

9. **Final result**: The square root of 605.16 is \(24.6\).

### Part (ii): Finding the square root of 0.000729

1. **Pair the digits**: Start pairing the digits from the decimal point. For 0.000729, we pair as (00)(00)(72)(9).

2. **Find the largest square**: The largest perfect square less than or equal to 0 is 0. Write 0 above the first pair (00).

\[ \begin{array}{c|c} 0 & 0 \\ \hline 0 & 00 \\ \end{array} \]

3. **Bring down**: Bring down the next pair (00) to keep it 0.

4. **Double the quotient**: Double the number above (0) to get 0. Now we need to find a digit \(x\) such that \(0x \times x\) is less than or equal to 72.

5. **Trial and error**:
- If \(x = 8\): \(08 \times 8 = 64\) (valid)
- Write 8 above the pair (72).

\[ \begin{array}{c|c} 0 & 0 & 8 \\ \hline 0 & 00 & 72 \\ 0 & 64 \\ \end{array} \]

6. **Subtract again**: Subtract 64 from 72, which gives 8. Bring down the next pair (29) to make it 829.

7. **Double the new quotient**: Double the number above (08) to get 16. Now we need to find a digit \(y\) such that \(16y \times y\) is less than or equal to 829.

8. **Trial and error**:
- If \(y = 5\): \(165 \times 5 = 825\) (valid)

Write 5 above the pair (29).

\[ \begin{array}{c|c} 0 & 0 & 8 & 5 \\ \hline 0 & 00 & 72 & 29 \\ 0 & 64 & 825 \\ \end{array} \]

9. **Final result**: The square root of 0.000729 is \(0.027\).

### Summary of Results
- The square root of 605.16 is **24.6**.
- The square root of 0.000729 is **0.027**.