The number of prime factors of `6^(10) xx 7^(17) xx 55^(27)` is:

Options

A. 54
B. 64
C. 81
D. 91

Answer

To find the number of prime factors of the expression \(6^{10} \times 7^{17} \times 55^{27}\), we will break down each component into its prime factors and then count the total number of prime factors.

### Step-by-step Solution:

1. **Break down each term into prime factors**:
- The prime factorization of \(6\) is \(2 \times 3\).
- The prime factorization of \(7\) is \(7\) (it is already a prime number).
- The prime factorization of \(55\) is \(5 \times 11\).

Thus, we can rewrite the expression:
\[ 6^{10} = (2 \times 3)^{10} = 2^{10} \times 3^{10} \]
\[ 7^{17} = 7^{17} \]
\[ 55^{27} = (5 \times 11)^{27} = 5^{27} \times 11^{27} \]

Combining these, we have:
\[ 6^{10} \times 7^{17} \times 55^{27} = 2^{10} \times 3^{10} \times 7^{17} \times 5^{27} \times 11^{27} \]

2. **List all the prime factors**:
The prime factors from the expression are:
- \(2\) with exponent \(10\)
- \(3\) with exponent \(10\)
- \(5\) with exponent \(27\)
- \(7\) with exponent \(17\)
- \(11\) with exponent \(27\)

3. **Count the distinct prime factors**:
The distinct prime factors are \(2\), \(3\), \(5\), \(7\), and \(11\).

Therefore, the number of distinct prime factors is:
\[ 5 \]

4. **Count the total number of prime factors**:
To find the total number of prime factors (counting multiplicities), we add the exponents:
\[ 10 + 10 + 27 + 17 + 27 = 91 \]

### Final Answer:
The total number of prime factors of \(6^{10} \times 7^{17} \times 55^{27}\) is **91**.

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