Show that 9261 is a perfect cube.

Answer

To show that 9261 is a perfect cube, we will perform the prime factorization of the number and check if all the prime factors can be grouped into triplets.

### Step-by-Step Solution:

1. **Start with the number 9261.**
- We need to factor this number into its prime components.

2. **Divide by the smallest prime number, which is 3.**
- \( 9261 \div 3 = 3087 \)
- So, we have one factor of 3.

3. **Continue dividing by 3.**
- \( 3087 \div 3 = 1029 \)
- Now we have two factors of 3.

4. **Divide 1029 by 3 again.**
- \( 1029 \div 3 = 343 \)
- Now we have three factors of 3.

5. **Next, we factor 343.**
- Since 343 is not divisible by 3, we try the next prime number, which is 7.
- \( 343 \div 7 = 49 \)
- Now we have one factor of 7.

6. **Continue factoring 49.**
- \( 49 \div 7 = 7 \)
- Now we have two factors of 7.

7. **Finally, divide 7 by 7.**
- \( 7 \div 7 = 1 \)
- Now we have three factors of 7.

8. **Collect all the prime factors.**
- From our divisions, we have:
- \( 9261 = 3 \times 3 \times 3 \times 7 \times 7 \times 7 \)
- This can be expressed as:
- \( 9261 = 3^3 \times 7^3 \)

9. **Check for triplets.**
- Both prime factors (3 and 7) appear in groups of three (triplets).

10. **Conclusion:**
- Since all the prime factors of 9261 can be grouped into triplets, we conclude that 9261 is a perfect cube.

### Final Statement:
Thus, we can say that \( 9261 = (3 \times 7)^3 = 21^3 \), confirming that 9261 is indeed a perfect cube.

---