Find the number of prime factors in `30^(7)xx22^(5)xx34^(12)xx12^(15)`

Options

A. 70
B. 65
C. 29
D. 69

Answer

To find the number of prime factors in the expression \(30^7 \times 22^5 \times 34^{12} \times 12^{15}\), we will follow these steps:

### Step 1: Factor each number into its prime factors.
- **30** can be factored as \(2^1 \times 3^1 \times 5^1\).
- **22** can be factored as \(2^1 \times 11^1\).
- **34** can be factored as \(2^1 \times 17^1\).
- **12** can be factored as \(2^2 \times 3^1\).

### Step 2: Substitute the prime factorization into the expression.
Now we can rewrite the expression using the prime factorizations:
\[ (2^1 \times 3^1 \times 5^1)^7 \times (2^1 \times 11^1)^5 \times (2^1 \times 17^1)^{12} \times (2^2 \times 3^1)^{15} \]

### Step 3: Apply the power to each factor.
Now we apply the powers:
\[ = 2^{7} \times 3^{7} \times 5^{7} \times 2^{5} \times 11^{5} \times 2^{12} \times 17^{12} \times 2^{30} \times 3^{15} \]

### Step 4: Combine the powers of the same base.
Next, we combine the powers of the same base:
- For \(2\):
\[ 7 + 5 + 12 + 30 = 54 \]
- For \(3\):
\[ 7 + 15 = 22 \]
- For \(5\):
\[ 7 \]
- For \(11\):
\[ 5 \]
- For \(17\):
\[ 12 \]

### Step 5: Write the expression in terms of prime factors.
Now we can express the entire expression in terms of its prime factors:
\[ = 2^{54} \times 3^{22} \times 5^{7} \times 11^{5} \times 17^{12} \]

### Step 6: Count the number of prime factors.
The total number of prime factors is the sum of the exponents:
\[ 54 + 22 + 7 + 5 + 12 = 100 \]

### Final Answer:
Thus, the number of prime factors in the expression \(30^7 \times 22^5 \times 34^{12} \times 12^{15}\) is **100**.

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