The total number of prime factors in `4^(10)xx7^(3)xx16^(2)xx11xx10^(2)` is

Options

A. `34`
B. `35`
C. `36`
D. `37`

Answer

To find the total number of prime factors in the expression \( 4^{10} \times 7^{3} \times 16^{2} \times 11 \times 10^{2} \), we will first factor each term into its prime factors.

### Step 1: Factor each term into prime factors

1. **Factor \( 4^{10} \)**:
\[ 4 = 2^2 \implies 4^{10} = (2^2)^{10} = 2^{20} \]

2. **Factor \( 7^{3} \)**:
\[ 7^{3} \text{ is already a prime factor.} \]

3. **Factor \( 16^{2} \)**:
\[ 16 = 2^4 \implies 16^{2} = (2^4)^{2} = 2^{8} \]

4. **Factor \( 11 \)**:
\[ 11 \text{ is already a prime factor.} \]

5. **Factor \( 10^{2} \)**:
\[ 10 = 2 \times 5 \implies 10^{2} = (2 \times 5)^{2} = 2^{2} \times 5^{2} \]

### Step 2: Combine all the prime factors

Now, we will combine all the prime factors together:
\[ 4^{10} \times 7^{3} \times 16^{2} \times 11 \times 10^{2} = 2^{20} \times 7^{3} \times 2^{8} \times 11 \times (2^{2} \times 5^{2}) \]

### Step 3: Simplify the expression

Combine the powers of \( 2 \):
\[ 2^{20} \times 2^{8} \times 2^{2} = 2^{(20 + 8 + 2)} = 2^{30} \]

Now, we can rewrite the entire expression:
\[ 2^{30} \times 7^{3} \times 11 \times 5^{2} \]

### Step 4: Identify the total number of prime factors

The prime factorization of the expression is:
\[ 2^{30}, 7^{3}, 11^{1}, 5^{2} \]

To find the total number of prime factors, we add the exponents of each prime factor:
- For \( 2^{30} \): 30 factors
- For \( 7^{3} \): 3 factors
- For \( 11^{1} \): 1 factor
- For \( 5^{2} \): 2 factors

### Total number of prime factors:
\[ 30 + 3 + 1 + 2 = 36 \]

Thus, the total number of prime factors in the expression \( 4^{10} \times 7^{3} \times 16^{2} \times 11 \times 10^{2} \) is **36**.

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