How many prime factors are there in the expression `(12)^(43)xx(34)^(48)xx(2)^(57)?`

Options

A. 282
B. 237
C. 142
D. 61

Answer

To find the number of prime factors in the expression \( (12)^{43} \times (34)^{48} \times (2)^{57} \), we will follow these steps:

### Step 1: Factorize each base into its prime factors.
- The prime factorization of \( 12 \) is \( 3 \times 2^2 \).
- The prime factorization of \( 34 \) is \( 17 \times 2 \).
- The prime factorization of \( 2 \) is \( 2 \).

### Step 2: Rewrite the expression using the prime factorizations.
Now we can rewrite the expression:
\[ (12)^{43} = (3 \times 2^2)^{43} = 3^{43} \times (2^2)^{43} = 3^{43} \times 2^{86} \]
\[ (34)^{48} = (17 \times 2)^{48} = 17^{48} \times 2^{48} \]
\[ (2)^{57} = 2^{57} \]

### Step 3: Combine all the prime factors.
Now we combine all the prime factors:
\[ 3^{43} \times 2^{86} \times 17^{48} \times 2^{48} \times 2^{57} \]

### Step 4: Combine the powers of the same base.
For the base \( 2 \):
\[ 2^{86} \times 2^{48} \times 2^{57} = 2^{(86 + 48 + 57)} = 2^{191} \]

So, the expression simplifies to:
\[ 3^{43} \times 17^{48} \times 2^{191} \]

### Step 5: Count the number of distinct prime factors.
The distinct prime factors in the expression are \( 3 \), \( 17 \), and \( 2 \).

### Step 6: Conclusion
Thus, the total number of prime factors (counting multiplicities) is:
\[ 43 + 48 + 191 = 282 \]

### Final Answer
The number of prime factors in the expression \( (12)^{43} \times (34)^{48} \times (2)^{57} \) is **282**.

---