The number of prime factors in the expression `6^4 xx 8^6 xx 10^8 xx 12^(10)` is:

Options

A. 48
B. 64
C. 72
D. 80

Answer

To find the number of prime factors in the expression \(6^4 \times 8^6 \times 10^8 \times 12^{10}\), we will first break down each term into its prime factors and then combine them.

### Step 1: Break down each number into prime factors
1. **For \(6\)**:
\[ 6 = 2^1 \times 3^1 \]
Therefore,
\[ 6^4 = (2^1 \times 3^1)^4 = 2^4 \times 3^4 \]

2. **For \(8\)**:
\[ 8 = 2^3 \]
Therefore,
\[ 8^6 = (2^3)^6 = 2^{18} \]

3. **For \(10\)**:
\[ 10 = 2^1 \times 5^1 \]
Therefore,
\[ 10^8 = (2^1 \times 5^1)^8 = 2^8 \times 5^8 \]

4. **For \(12\)**:
\[ 12 = 2^2 \times 3^1 \]
Therefore,
\[ 12^{10} = (2^2 \times 3^1)^{10} = 2^{20} \times 3^{10} \]

### Step 2: Combine all the prime factors
Now we can combine all the prime factors from each term:
\[ 6^4 \times 8^6 \times 10^8 \times 12^{10} = (2^4 \times 3^4) \times (2^{18}) \times (2^8 \times 5^8) \times (2^{20} \times 3^{10}) \]

### Step 3: Add the powers of the same base
Now we will add the powers of the same base:
- For base \(2\):
\[ 4 + 18 + 8 + 20 = 50 \]
- For base \(3\):
\[ 4 + 10 = 14 \]
- For base \(5\):
\[ 8 \]

### Step 4: Write the final expression
The expression can now be written as:
\[ 2^{50} \times 3^{14} \times 5^{8} \]

### Step 5: Count the total number of prime factors
To find the total number of prime factors, we simply add the powers together:
\[ 50 + 14 + 8 = 72 \]

Thus, the total number of prime factors in the expression \(6^4 \times 8^6 \times 10^8 \times 12^{10}\) is **72**.

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