Calculate the total number of prime factors in the expression : `(9)^(11)xx(5)^(7)xx(7)^(5)xx(3)^(2)xx(17)^(2)`

Options

A. 35
B. 36
C. 37
D. 38

Answer

To find the total number of prime factors in the expression \( (9)^{11} \times (5)^{7} \times (7)^{5} \times (3)^{2} \times (17)^{2} \), we can follow these steps:

### Step 1: Rewrite the expression using prime factors
First, we need to express \( 9 \) in terms of its prime factors.
Since \( 9 = 3^2 \), we can rewrite \( (9)^{11} \) as follows:
\[ (9)^{11} = (3^2)^{11} = 3^{22} \]

### Step 2: Substitute back into the expression
Now we can substitute \( 3^{22} \) back into the original expression:
\[ 3^{22} \times 5^{7} \times 7^{5} \times 3^{2} \times 17^{2} \]

### Step 3: Combine like terms
Next, we combine the powers of \( 3 \):
\[ 3^{22} \times 3^{2} = 3^{22 + 2} = 3^{24} \]
So the expression now becomes:
\[ 3^{24} \times 5^{7} \times 7^{5} \times 17^{2} \]

### Step 4: Identify the prime factors and their powers
The prime factors in the expression are:
- \( 3^{24} \)
- \( 5^{7} \)
- \( 7^{5} \)
- \( 17^{2} \)

### Step 5: Count the total number of prime factors
To find the total number of prime factors, we add the powers of each prime factor:
\[ 24 + 7 + 5 + 2 = 38 \]

Thus, the total number of prime factors in the expression is \( 38 \).

### Final Answer
The total number of prime factors is \( 38 \).

---