Find the total number of Prime factors in the given expression ?
`(30)^25xx(25)^51xx(12)^23`

Options

A. 249
B. 250
C. 255
D. 260

Answer

To find the total number of prime factors in the expression \( (30)^{25} \times (25)^{51} \times (12)^{23} \), we will break down each base into its prime factors and then calculate the total number of prime factors.

### Step 1: Prime Factorization of Each Base
1. **Prime Factorization of 30**:
\[ 30 = 2 \times 3 \times 5 \]
Therefore,
\[ (30)^{25} = (2 \times 3 \times 5)^{25} = 2^{25} \times 3^{25} \times 5^{25} \]

2. **Prime Factorization of 25**:
\[ 25 = 5^2 \]
Therefore,
\[ (25)^{51} = (5^2)^{51} = 5^{102} \]

3. **Prime Factorization of 12**:
\[ 12 = 3 \times 4 = 3 \times (2^2) = 3 \times 2^2 \]
Therefore,
\[ (12)^{23} = (3 \times 2^2)^{23} = 3^{23} \times 2^{46} \]

### Step 2: Combine All the Prime Factors
Now we combine all the prime factors from the three expressions:
\[ (30)^{25} \times (25)^{51} \times (12)^{23} = (2^{25} \times 3^{25} \times 5^{25}) \times (5^{102}) \times (3^{23} \times 2^{46}) \]

### Step 3: Add the Exponents of Each Prime Factor
Now we will sum the exponents for each prime factor:
- For \(2\):
\[ 25 + 46 = 71 \]
- For \(3\):
\[ 25 + 23 = 48 \]
- For \(5\):
\[ 25 + 102 = 127 \]

### Step 4: Total Number of Prime Factors
Now we can find the total number of prime factors by summing the individual contributions:
\[ 71 + 48 + 127 = 246 \]

Thus, the total number of prime factors in the expression \( (30)^{25} \times (25)^{51} \times (12)^{23} \) is **246**.

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