Is 297 a perfect cube ?

Answer

To determine if 297 is a perfect cube, we will follow these steps:

### Step 1: Factorize 297
We start by finding the prime factorization of 297. Since 297 is an odd number, we cannot divide it by 2. We can check for divisibility by 3.

- **Dividing by 3:**
\[ 297 \div 3 = 99 \]
- **Dividing 99 by 3 again:**
\[ 99 \div 3 = 33 \]
- **Dividing 33 by 3 again:**
\[ 33 \div 3 = 11 \]
- **11 is a prime number.**

So, the prime factorization of 297 is:
\[ 297 = 3^3 \times 11^1 \]

### Step 2: Analyze the Prime Factorization
In order for a number to be a perfect cube, all the prime factors must have exponents that are multiples of 3.

From our factorization:
- The exponent of 3 is 3 (which is a multiple of 3).
- The exponent of 11 is 1 (which is not a multiple of 3).

### Step 3: Conclusion
Since not all the prime factors have exponents that are multiples of 3, we conclude that 297 is not a perfect cube.

### Final Answer
**297 is not a perfect cube.**

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