Find the total number of Prime factors in the given expression ?
`(30)^(15)xx(22)^11xx(15)^24`

Options

A. 110
B. 115
C. 120
D. 125

Answer

To find the total number of prime factors in the expression \( (30)^{15} \times (22)^{11} \times (15)^{24} \), we will follow these steps:

### Step 1: Prime Factorization of Each Base
We start by finding the prime factorization of each base in the expression.

1. **For \( 30 \)**:
\[ 30 = 2^1 \times 3^1 \times 5^1 \]

2. **For \( 22 \)**:
\[ 22 = 2^1 \times 11^1 \]

3. **For \( 15 \)**:
\[ 15 = 3^1 \times 5^1 \]

### Step 2: Rewrite the Expression with Prime Factors
Now we can rewrite the entire expression using the prime factorizations we found.

\[ (30)^{15} = (2^1 \times 3^1 \times 5^1)^{15} = 2^{15} \times 3^{15} \times 5^{15} \]

\[ (22)^{11} = (2^1 \times 11^1)^{11} = 2^{11} \times 11^{11} \]

\[ (15)^{24} = (3^1 \times 5^1)^{24} = 3^{24} \times 5^{24} \]

### Step 3: Combine All Prime Factors
Now we combine all the prime factors from the rewritten expressions:

\[ = 2^{15} \times 3^{15} \times 5^{15} \times 2^{11} \times 11^{11} \times 3^{24} \times 5^{24} \]

### Step 4: Combine Like Terms
Next, we combine the like terms (the same bases):

1. For \( 2 \):
\[ 2^{15} \times 2^{11} = 2^{15 + 11} = 2^{26} \]

2. For \( 3 \):
\[ 3^{15} \times 3^{24} = 3^{15 + 24} = 3^{39} \]

3. For \( 5 \):
\[ 5^{15} \times 5^{24} = 5^{15 + 24} = 5^{39} \]

4. For \( 11 \):
\[ 11^{11} \]

So, the expression can be rewritten as:
\[ 2^{26} \times 3^{39} \times 5^{39} \times 11^{11} \]

### Step 5: Count the Total Number of Prime Factors
Now, we count the total number of prime factors by adding the exponents:

\[ 26 \text{ (for } 2\text{)} + 39 \text{ (for } 3\text{)} + 39 \text{ (for } 5\text{)} + 11 \text{ (for } 11\text{)} \]

Calculating this gives:
\[ 26 + 39 + 39 + 11 = 115 \]

### Final Answer
Thus, the total number of prime factors in the given expression is \( \boxed{115} \).

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