Find decimal form of `sqrt(23)` and `sqrt(24)` upto 3 decimal place.

Answer

To find the decimal form of \( \sqrt{23} \) and \( \sqrt{24} \) up to three decimal places, we can use the long division method. Here’s a step-by-step solution:

### Step 1: Calculate \( \sqrt{23} \)

1. **Identify the nearest perfect square**: The nearest perfect square less than 23 is \( 16 \) (which is \( 4^2 \)).
2. **Set up the long division**:
- Write \( 4.00 \) (since \( \sqrt{16} = 4 \)) and add two decimal places for further calculation.
3. **Subtract**:
- \( 23 - 16 = 7 \)
4. **Bring down two zeros**:
- Now we have \( 700 \).
5. **Find the next digit**:
- Double the current quotient \( 4 \) to get \( 8 \) and find a digit \( x \) such that \( 8x \times x \) is as close to \( 700 \) as possible.
- Trying \( 87 \): \( 87 \times 7 = 609 \) (this works).
6. **Subtract**:
- \( 700 - 609 = 91 \)
7. **Bring down two more zeros**:
- Now we have \( 9100 \).
8. **Find the next digit**:
- Double \( 87 \) to get \( 174 \) and find \( y \) such that \( 174y \times y \) is as close to \( 9100 \) as possible.
- Trying \( 9 \): \( 1749 \times 9 = 15741 \) (too high), trying \( 8 \): \( 1748 \times 8 = 13984 \) (too high), trying \( 7 \): \( 1747 \times 7 = 12229 \) (too high), trying \( 6 \): \( 1746 \times 6 = 10476 \) (too high), trying \( 5 \): \( 1745 \times 5 = 8725 \) (too high), trying \( 4 \): \( 1744 \times 4 = 6976 \) (this works).
9. **Subtract**:
- \( 9100 - 6976 = 2124 \)
10. **Bring down two more zeros**:
- Now we have \( 212400 \).
11. **Find the next digit**:
- Double \( 874 \) to get \( 1748 \) and find \( z \) such that \( 1748z \times z \) is as close to \( 212400 \) as possible.
- Trying \( 1 \): \( 17481 \times 1 = 17481 \) (this works).
12. **Final result**:
- After performing the calculations, we find that \( \sqrt{23} \approx 4.795 \).

### Step 2: Calculate \( \sqrt{24} \)

1. **Identify the nearest perfect square**: The nearest perfect square less than 24 is \( 16 \) (which is \( 4^2 \)).
2. **Set up the long division**:
- Write \( 4.00 \) (since \( \sqrt{16} = 4 \)) and add two decimal places.
3. **Subtract**:
- \( 24 - 16 = 8 \)
4. **Bring down two zeros**:
- Now we have \( 800 \).
5. **Find the next digit**:
- Double the current quotient \( 4 \) to get \( 8 \) and find a digit \( x \) such that \( 8x \times x \) is as close to \( 800 \) as possible.
- Trying \( 8 \): \( 88 \times 8 = 704 \) (this works).
6. **Subtract**:
- \( 800 - 704 = 96 \)
7. **Bring down two more zeros**:
- Now we have \( 9600 \).
8. **Find the next digit**:
- Double \( 88 \) to get \( 176 \) and find \( y \) such that \( 176y \times y \) is as close to \( 9600 \) as possible.
- Trying \( 5 \): \( 1765 \times 5 = 8825 \) (this works).
9. **Subtract**:
- \( 9600 - 8825 = 775 \)
10. **Bring down two more zeros**:
- Now we have \( 77500 \).
11. **Find the next digit**:
- Double \( 885 \) to get \( 1770 \) and find \( z \) such that \( 1770z \times z \) is as close to \( 77500 \) as possible.
- Trying \( 4 \): \( 17704 \times 4 = 70816 \) (this works).
12. **Final result**:
- After performing the calculations, we find that \( \sqrt{24} \approx 4.898 \).

### Final Answers:
- \( \sqrt{23} \approx 4.795 \)
- \( \sqrt{24} \approx 4.898 \)