The total number of prime factors which are contained in `(30)^6` is

Options

A. 15
B. 16
C. 17
D. 18

Answer

To find the total number of prime factors contained in \( (30)^6 \), we can follow these steps:

### Step 1: Factorize the base number
First, we need to factorize the number 30 into its prime factors.

\[ 30 = 2 \times 3 \times 5 \]

### Step 2: Raise the prime factors to the power
Next, we raise each of the prime factors to the power of 6, since we are considering \( (30)^6 \).

\[ (30)^6 = (2 \times 3 \times 5)^6 \]

### Step 3: Apply the power to each factor
Using the property of exponents, we can distribute the exponent 6 to each of the prime factors:

\[ (2^1 \times 3^1 \times 5^1)^6 = 2^{6} \times 3^{6} \times 5^{6} \]

### Step 4: Count the total number of prime factors
Now, we count the total number of prime factors. Each prime factor contributes its exponent to the total count. Since all prime factors are raised to the power of 6, we can calculate the total number of prime factors:

\[ \text{Total number of prime factors} = 6 + 6 + 6 = 18 \]

### Final Answer
Thus, the total number of prime factors contained in \( (30)^6 \) is:

\[ \boxed{18} \]

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