if `x in N`, then the value of x satisfying the equation `5^x*(8^(x-1))^(1/x)=500` is divisible by

Options

A. 2
B. 4
C. 3
D. 5

Explanation

To solve the equation \( 5^x \cdot (8^{x-1})^{1/x} = 500 \) for \( x \in \mathbb{N} \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ 5^x \cdot (8^{x-1})^{1/x} = 500 \]
We can rewrite \( 8 \) as \( 2^3 \), so:
\[ 8^{x-1} = (2^3)^{x-1} = 2^{3(x-1)} \]
Thus, we have:
\[ (8^{x-1})^{1/x} = (2^{3(x-1)})^{1/x} = 2^{\frac{3(x-1)}{x}} = 2^{3 - \frac{3}{x}} \]
Now, substituting this back into the equation gives us:
\[ 5^x \cdot 2^{3 - \frac{3}{x}} = 500 \]

### Step 2: Express 500 in terms of prime factors
Next, we express \( 500 \) in terms of its prime factors:
\[ 500 = 5^3 \cdot 2^2 \]
Now, we can rewrite our equation as:
\[ 5^x \cdot 2^{3 - \frac{3}{x}} = 5^3 \cdot 2^2 \]

### Step 3: Compare the exponents
Since the bases are the same, we can equate the exponents for both bases.

1. For base \( 5 \):
\[ x = 3 \]

2. For base \( 2 \):
\[ 3 - \frac{3}{x} = 2 \]
Rearranging gives:
\[ 3 - 2 = \frac{3}{x} \implies 1 = \frac{3}{x} \implies x = 3 \]

### Step 4: Conclusion
Both comparisons give us \( x = 3 \). Since \( x \) must be a natural number, we conclude that the only solution is:
\[ x = 3 \]

### Step 5: Check divisibility
Now, we need to determine what \( x \) is divisible by. Since \( x = 3 \), we can say:
\[ 3 \text{ is divisible by } 3 \]

### Final Answer
The value of \( x \) satisfying the equation is \( 3 \), and it is divisible by \( 3 \).

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