Calculate the total number of prime factors in the expression : `(4)^(11)xx(5)^(5)xx(3)^(2)xx(13)^(2)`

Options

A. 30
B. 31
C. 33
D. 32

Answer

To calculate the total number of prime factors in the expression \( (4)^{11} \times (5)^{5} \times (3)^{2} \times (13)^{2} \), we will follow these steps:

### Step 1: Identify the prime factors in the expression
The expression consists of the following terms:
- \( 4^{11} \)
- \( 5^{5} \)
- \( 3^{2} \)
- \( 13^{2} \)

Now, we need to express \( 4 \) in terms of its prime factors:
- \( 4 = 2^2 \)

Thus, we can rewrite \( 4^{11} \) as:
\[ (2^2)^{11} = 2^{22} \]

Now, the expression can be rewritten using only prime factors:
\[ 2^{22} \times 5^{5} \times 3^{2} \times 13^{2} \]

### Step 2: List all the prime factors
From the rewritten expression, we can identify the prime factors:
- \( 2 \) (from \( 2^{22} \))
- \( 3 \) (from \( 3^{2} \))
- \( 5 \) (from \( 5^{5} \))
- \( 13 \) (from \( 13^{2} \))

### Step 3: Count the total number of prime factors
To find the total number of prime factors, we need to consider the powers of each prime factor:
- The power of \( 2 \) is \( 22 \)
- The power of \( 3 \) is \( 2 \)
- The power of \( 5 \) is \( 5 \)
- The power of \( 13 \) is \( 2 \)

Now, we sum these powers:
\[ 22 + 5 + 2 + 2 = 31 \]

### Final Answer
The total number of prime factors in the expression \( (4)^{11} \times (5)^{5} \times (3)^{2} \times (13)^{2} \) is \( 31 \).

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