Is 196 a perfect cube?

Answer

To determine if 196 is a perfect cube, we need to follow these steps:

### Step 1: Find the Prime Factors of 196
To find the prime factors, we can start dividing 196 by the smallest prime number, which is 2.

- 196 ÷ 2 = 98
- 98 ÷ 2 = 49
- 49 ÷ 7 = 7
- 7 ÷ 7 = 1

So, the prime factorization of 196 is:
\[ 196 = 2^2 \times 7^2 \]

### Step 2: Analyze the Prime Factors
Next, we need to check if the prime factors can form triplets. For a number to be a perfect cube, each prime factor must appear in multiples of three.

From our factorization:
- The prime factor 2 appears 2 times (which is \(2^2\)).
- The prime factor 7 appears 2 times (which is \(7^2\)).

### Step 3: Conclusion
Since neither of the prime factors (2 and 7) appears in multiples of three, we conclude that 196 does not form triplets.

Thus, **196 is not a perfect cube**.

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