Calculate the total numbers of prime factors in the expression `(9) ^(11) xx (5)^(7) xx (7) ^(5) xx (3) (2) (17) ^(2) `
(a)35
(b)36
(c)37
(d)38

Options

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

Answer

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

### Step 1: Rewrite the expression with prime bases
First, we need to express all numbers in the expression using their prime factors.

- \( 9 = 3^2 \), so \( (9)^{11} = (3^2)^{11} = 3^{22} \).

Now, we can rewrite the entire expression:
\[ (3^{22}) \times (5^{7}) \times (7^{5}) \times (3^{1}) \times (2^{1}) \times (17^{2}) \]

### Step 2: Combine like bases
Next, we combine the powers of the same base.

- For base \( 3 \): \( 3^{22} \times 3^{1} = 3^{22 + 1} = 3^{23} \).

Now the expression becomes:
\[ 3^{23} \times 5^{7} \times 7^{5} \times 2^{1} \times 17^{2} \]

### Step 3: Identify the prime factors and their powers
Now we list the prime factors and their respective powers:
- \( 3^{23} \)
- \( 5^{7} \)
- \( 7^{5} \)
- \( 2^{1} \)
- \( 17^{2} \)

### Step 4: Calculate the total number of prime factors
To find the total number of prime factors, we add the powers of each prime factor:
- The power of \( 3 \) is \( 23 \).
- The power of \( 5 \) is \( 7 \).
- The power of \( 7 \) is \( 5 \).
- The power of \( 2 \) is \( 1 \).
- The power of \( 17 \) is \( 2 \).

Now, we calculate:
\[ 23 + 7 + 5 + 1 + 2 = 38 \]

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