What will be the sum of exponents of prime factors in the prime factorisation of 196?

Options

A. 2
B. 4
C. 6
D. 5

Answer

To find the sum of the exponents of the prime factors in the prime factorization of 196, we will follow these steps:

### Step 1: Find the prime factorization of 196.
To do this, we can start by dividing 196 by the smallest prime number, which is 2.

- **196 ÷ 2 = 98** (So, 2 is a prime factor)
- **98 ÷ 2 = 49** (2 is still a prime factor)
- Now, we need to factor 49. The smallest prime number that can divide 49 is 7.
- **49 ÷ 7 = 7** (So, 7 is also a prime factor)
- **7 ÷ 7 = 1** (This means we have completely factored 49)

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

### Step 2: Identify the exponents of the prime factors.
From the prime factorization \( 2^2 \times 7^2 \), we can see that:
- The exponent of the prime factor 2 is 2.
- The exponent of the prime factor 7 is 2.

### Step 3: Calculate the sum of the exponents.
Now, we add the exponents together:
\[ 2 + 2 = 4 \]

### Final Answer:
The sum of the exponents of the prime factors in the prime factorization of 196 is **4**.

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