Which of the following number are perfect cubes? In case of perfect cube, find the whose cubes is the given number.
(i) 125 (ii) 243 (iii) 343 (iv) 256 (v) 8000 (vi) 9261 (vii) 5324 (viii) 3375

Answer

To determine which of the given numbers are perfect cubes and to find the cube root for those that are, we will analyze each number step by step.

### Step-by-Step Solution:

1. **Check 125:**
- Factors: \( 125 = 5 \times 5 \times 5 = 5^3 \)
- Conclusion: **125 is a perfect cube.**
- Cube root: \( \sqrt[3]{125} = 5 \)

2. **Check 243:**
- Factors: \( 243 = 3 \times 3 \times 3 \times 3 \times 3 = 3^5 \)
- Grouping: Cannot form groups of three.
- Conclusion: **243 is not a perfect cube.**

3. **Check 343:**
- Factors: \( 343 = 7 \times 7 \times 7 = 7^3 \)
- Conclusion: **343 is a perfect cube.**
- Cube root: \( \sqrt[3]{343} = 7 \)

4. **Check 256:**
- Factors: \( 256 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^8 \)
- Grouping: Cannot form groups of three.
- Conclusion: **256 is not a perfect cube.**

5. **Check 8000:**
- Factors: \( 8000 = 20 \times 20 \times 20 = 20^3 \)
- Conclusion: **8000 is a perfect cube.**
- Cube root: \( \sqrt[3]{8000} = 20 \)

6. **Check 9261:**
- Factors: \( 9261 = 21 \times 21 \times 21 = 21^3 \)
- Conclusion: **9261 is a perfect cube.**
- Cube root: \( \sqrt[3]{9261} = 21 \)

7. **Check 5324:**
- Factors: \( 5324 = 2 \times 2 \times 11 \times 11 \times 11 \)
- Grouping: Cannot form groups of three.
- Conclusion: **5324 is not a perfect cube.**

8. **Check 3375:**
- Factors: \( 3375 = 15 \times 15 \times 15 = 15^3 \)
- Conclusion: **3375 is a perfect cube.**
- Cube root: \( \sqrt[3]{3375} = 15 \)

### Summary of Results:
- Perfect cubes:
- 125 (Cube root: 5)
- 343 (Cube root: 7)
- 8000 (Cube root: 20)
- 9261 (Cube root: 21)
- 3375 (Cube root: 15)

- Not perfect cubes:
- 243
- 256
- 5324