The number of prime factors in
`((1)/(6))^(12)xx(8)^(25)xx((3)/(4))^(15)` is

Options

A. 33
B. 37
C. 52
D. None of these

Answer

To find the number of prime factors in the expression \(\left(\frac{1}{6}\right)^{12} \times (8)^{25} \times \left(\frac{3}{4}\right)^{15}\), we will simplify each component and then combine them.

### Step 1: Simplify each term

1. **Simplify \(\left(\frac{1}{6}\right)^{12}\)**:
\[ \left(\frac{1}{6}\right)^{12} = \frac{1^{12}}{6^{12}} = \frac{1}{(2 \times 3)^{12}} = \frac{1}{2^{12} \times 3^{12}} \]

2. **Simplify \((8)^{25}\)**:
\[ (8)^{25} = (2^3)^{25} = 2^{75} \]

3. **Simplify \(\left(\frac{3}{4}\right)^{15}\)**:
\[ \left(\frac{3}{4}\right)^{15} = \frac{3^{15}}{4^{15}} = \frac{3^{15}}{(2^2)^{15}} = \frac{3^{15}}{2^{30}} = 3^{15} \times 2^{-30} \]

### Step 2: Combine all terms

Now we combine all the simplified terms:
\[ \frac{1}{2^{12} \times 3^{12}} \times 2^{75} \times (3^{15} \times 2^{-30}) \]

### Step 3: Combine the powers of the same bases

1. For base \(2\):
\[ 2^{-12} \times 2^{75} \times 2^{-30} = 2^{-12 + 75 - 30} = 2^{33} \]

2. For base \(3\):
\[ 3^{-12} \times 3^{15} = 3^{-12 + 15} = 3^{3} \]

### Step 4: Write the final expression

Now we can write the entire expression as:
\[ 2^{33} \times 3^{3} \]

### Step 5: Count the prime factors

The prime factors in the expression \(2^{33} \times 3^{3}\) are:
- The prime factor \(2\) with an exponent of \(33\)
- The prime factor \(3\) with an exponent of \(3\)

### Conclusion

The total number of prime factors (counting multiplicities) is:
\[ 33 + 3 = 36 \]

Thus, the number of prime factors in the expression is **36**.

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