Find the square root of :
(i) `3`, correct to three places of decimal.
(ii) `0.07688`, correct to two places of decimal.

Answer

To find the square roots of the given numbers step by step, we will use the long division method for both parts of the question.

### Part (i): Find the square root of 3, correct to three decimal places.

**Step 1:** Identify the nearest perfect square.
- The nearest perfect square to 3 is 1 (since \(1^2 = 1\)).

**Step 2:** Set up the long division.
- We write 3 as 3.000000 (adding decimal places for precision).
- We will find the square root of 3.000000.

**Step 3:** Start the long division process.
- The first pair is 3. The largest number whose square is less than or equal to 3 is 1.
- Write 1 above the line and subtract \(1^2 = 1\) from 3, leaving us with 2.

**Step 4:** Bring down the next pair of zeros.
- Now we have 200. Double the number above the line (1) to get 2. Write it down.

**Step 5:** Find the next digit.
- We need to find a digit \(x\) such that \(2x \cdot x\) is as close to 200 as possible.
- Trying \(x = 7\): \(27 \cdot 7 = 189\).
- Subtract 189 from 200, which leaves us with 11.

**Step 6:** Bring down the next pair of zeros.
- Now we have 1100. Double the current quotient (17) to get 34.

**Step 7:** Find the next digit.
- We need to find \(y\) such that \(34y \cdot y\) is as close to 1100 as possible.
- Trying \(y = 3\): \(343 \cdot 3 = 1029\).
- Subtract 1029 from 1100, which leaves us with 71.

**Step 8:** Bring down the next pair of zeros.
- Now we have 7100. Double the current quotient (173) to get 346.

**Step 9:** Find the next digit.
- We need to find \(z\) such that \(346z \cdot z\) is as close to 7100 as possible.
- Trying \(z = 2\): \(3462 \cdot 2 = 6924\).
- Subtract 6924 from 7100, which leaves us with 176.

**Step 10:** Finalize the result.
- After repeating this process, we find that the square root of 3 is approximately \(1.732\) correct to three decimal places.

### Part (ii): Find the square root of 0.07688, correct to two decimal places.

**Step 1:** Adjust the number for long division.
- We can rewrite \(0.07688\) as \(0.0768800\) (adding zeros to make pairs).

**Step 2:** Pair the digits.
- The pairs are (0.07)(68)(80)(00).

**Step 3:** Start the long division.
- The first pair is 0.07. The largest perfect square less than 0.07 is \(0.04\) (which is \(0.2^2\)).
- Write 0.2 above the line and subtract \(0.04\) from \(0.07\) to get \(0.03\).

**Step 4:** Bring down the next pair (68).
- Now we have 368. Double the current quotient (0.2) to get 0.4.

**Step 5:** Find the next digit.
- We need to find \(x\) such that \(0.4x \cdot x\) is as close to 368 as possible.
- Trying \(x = 7\): \(0.47 \cdot 7 = 3.29\).
- Subtract \(3.29\) from \(368\) to get \(39\).

**Step 6:** Bring down the next pair (80).
- Now we have 3980. Double the current quotient (0.27) to get 0.54.

**Step 7:** Find the next digit.
- We need to find \(y\) such that \(0.54y \cdot y\) is as close to 3980 as possible.
- Trying \(y = 2\): \(0.542 \cdot 2 = 1.088\).
- Subtract \(1.088\) from \(3980\) to get the remainder.

**Step 8:** Finalize the result.
- After repeating this process, we find that the square root of \(0.07688\) is approximately \(0.27\) correct to two decimal places.

### Summary of Results:
- (i) The square root of \(3\) is \(1.732\).
- (ii) The square root of \(0.07688\) is \(0.27\).