is 1296 a perfect cube or not ?

Answer

To determine whether 1296 is a perfect cube, we will follow these steps:

### Step 1: Prime Factorization of 1296
We start by finding the prime factors of 1296 through division.

1. Divide 1296 by 2:
- 1296 ÷ 2 = 648
2. Divide 648 by 2:
- 648 ÷ 2 = 324
3. Divide 324 by 2:
- 324 ÷ 2 = 162
4. Divide 162 by 2:
- 162 ÷ 2 = 81
5. Now, 81 is not divisible by 2, so we divide by 3:
- 81 ÷ 3 = 27
6. Divide 27 by 3:
- 27 ÷ 3 = 9
7. Divide 9 by 3:
- 9 ÷ 3 = 3
8. Finally, divide 3 by 3:
- 3 ÷ 3 = 1

### Step 2: Collecting the Prime Factors
From the above divisions, we can collect the prime factors:
- 1296 = 2^4 × 3^4

### Step 3: Checking the Powers of Prime Factors
To determine if 1296 is a perfect cube, we need to check the powers of the prime factors:
- The power of 2 is 4.
- The power of 3 is 4.

### Step 4: Checking for Multiples of 3
For a number to be a perfect cube, the powers of all prime factors must be multiples of 3:
- 4 is not a multiple of 3.

### Conclusion
Since the powers of the prime factors (4 and 4) are not multiples of 3, we conclude that:
- **1296 is not a perfect cube.**

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