Evaluate `(i)sqrt(4356)`
`(ii)sqrt (11025)`

Answer

To evaluate the square roots of the given numbers, we will use the division method step by step.

### (i) Evaluate \(\sqrt{4356}\)

**Step 1: Pair the digits from right to left.**
- The number is 4356. We pair the digits as (43)(56).

**Step 2: Find the largest square less than or equal to the first pair (43).**
- The largest square less than 43 is \(6^2 = 36\).
- Write down 6 as the first digit of the square root.

**Step 3: Subtract and bring down the next pair.**
- Subtract \(36\) from \(43\) to get \(7\).
- Bring down the next pair (56) to make it \(756\).

**Step 4: Double the current quotient (6) and find the next digit.**
- Double \(6\) to get \(12\).
- We need to find a digit \(x\) such that \(12x \times x\) is less than or equal to \(756\).
- Testing \(x = 6\): \(126 \times 6 = 756\) (which fits perfectly).

**Step 5: Subtract and conclude.**
- Subtract \(756\) from \(756\) to get \(0\).
- Thus, \(\sqrt{4356} = 66\).

### (ii) Evaluate \(\sqrt{11025}\)

**Step 1: Pair the digits from right to left.**
- The number is 11025. We pair the digits as (11)(02)(25).

**Step 2: Find the largest square less than or equal to the first pair (11).**
- The largest square less than 11 is \(3^2 = 9\).
- Write down 3 as the first digit of the square root.

**Step 3: Subtract and bring down the next pair.**
- Subtract \(9\) from \(11\) to get \(2\).
- Bring down the next pair (02) to make it \(202\).

**Step 4: Double the current quotient (3) and find the next digit.**
- Double \(3\) to get \(6\).
- We need to find a digit \(y\) such that \(6y \times y\) is less than or equal to \(202\).
- Testing \(y = 3\): \(63 \times 3 = 189\) (which fits).

**Step 5: Subtract and bring down the next pair.**
- Subtract \(189\) from \(202\) to get \(13\).
- Bring down the next pair (25) to make it \(1325\).

**Step 6: Double the current quotient (33) and find the next digit.**
- Double \(33\) to get \(66\).
- We need to find a digit \(z\) such that \(66z \times z\) is less than or equal to \(1325\).
- Testing \(z = 5\): \(665 \times 5 = 3325\) (too large).
- Testing \(z = 4\): \(664 \times 4 = 2656\) (too large).
- Testing \(z = 3\): \(663 \times 3 = 1989\) (too large).
- Testing \(z = 2\): \(662 \times 2 = 1324\) (fits perfectly).

**Step 7: Subtract and conclude.**
- Subtract \(1324\) from \(1325\) to get \(1\).
- Thus, \(\sqrt{11025} = 105\).

### Final Answers:
- \(\sqrt{4356} = 66\)
- \(\sqrt{11025} = 105\)