Number of prime factor of
`(216)^(3//5)xx(2500)^(2//5)xx(300)^(1//5)` is

Options

A. 3.5
B. 4.5
C. 6
D. 7

Answer

To solve the problem of finding the number of prime factors of the expression \((216)^{3/5} \times (2500)^{2/5} \times (300)^{1/5}\), we will follow these steps:

### Step 1: Prime Factorization of Each Number
First, we need to find the prime factorization of each number involved in the expression.

1. **Prime Factorization of 216**:
\[ 216 = 6^3 = (2 \times 3)^3 = 2^3 \times 3^3 \]

2. **Prime Factorization of 2500**:
\[ 2500 = 25^2 = (5^2)^2 = 5^4 \]

3. **Prime Factorization of 300**:
\[ 300 = 3 \times 100 = 3 \times (10^2) = 3 \times (2 \times 5)^2 = 3^1 \times 2^2 \times 5^2 \]

### Step 2: Substitute Prime Factorizations into the Expression
Now, we will substitute the prime factorizations back into the original expression:
\[ (216)^{3/5} = (2^3 \times 3^3)^{3/5} = 2^{(3 \cdot 3/5)} \times 3^{(3 \cdot 3/5)} = 2^{9/5} \times 3^{9/5} \]
\[ (2500)^{2/5} = (5^4)^{2/5} = 5^{(4 \cdot 2/5)} = 5^{8/5} \]
\[ (300)^{1/5} = (3^1 \times 2^2 \times 5^2)^{1/5} = 3^{(1/5)} \times 2^{(2/5)} \times 5^{(2/5)} \]

### Step 3: Combine the Expressions
Now we will combine all these results:
\[ (216)^{3/5} \times (2500)^{2/5} \times (300)^{1/5} = (2^{9/5} \times 3^{9/5}) \times (5^{8/5}) \times (3^{1/5} \times 2^{2/5} \times 5^{2/5}) \]

### Step 4: Combine Like Bases
Now we will combine the powers of the same bases:
\[ = 2^{(9/5 + 2/5)} \times 3^{(9/5 + 1/5)} \times 5^{(8/5 + 2/5)} \]
Calculating the exponents:
- For \(2\): \(9/5 + 2/5 = 11/5\)
- For \(3\): \(9/5 + 1/5 = 10/5 = 2\)
- For \(5\): \(8/5 + 2/5 = 10/5 = 2\)

Thus, we have:
\[ = 2^{11/5} \times 3^{2} \times 5^{2} \]

### Step 5: Determine the Number of Prime Factors
The number of prime factors is determined by the formula:
\[ \text{Number of prime factors} = (e_1 + 1)(e_2 + 1)(e_3 + 1) \]
where \(e_1, e_2, e_3\) are the exponents of the prime factors.

In our case:
- For \(2^{11/5}\), the exponent is \(11/5\) (considering only integer part, which is 2)
- For \(3^{2}\), the exponent is \(2\)
- For \(5^{2}\), the exponent is \(2\)

Calculating:
\[ \text{Number of prime factors} = (2 + 1)(2 + 1)(2 + 1) = 3 \times 3 \times 3 = 27 \]

### Final Answer
The number of prime factors of the expression \((216)^{3/5} \times (2500)^{2/5} \times (300)^{1/5}\) is **27**.