If `m^(n).n^(m)=800,` then the value of `(n)/(m)` is (m `<` n)

Options

A. `(1)/(2)`
B. `(1)/(5)`
C. `(4)/(5)`
D. `(5)/(2)`

Answer

To solve the equation \( m^n \cdot n^m = 800 \) given that \( m < n \), we can follow these steps:

### Step 1: Factor the number 800
First, we need to factor 800 into its prime factors.

\[ 800 = 2^5 \cdot 5^2 \]

### Step 2: Set up the equation
Now, we can rewrite the equation using the prime factorization:

\[ m^n \cdot n^m = 2^5 \cdot 5^2 \]

### Step 3: Identify possible values for \( m \) and \( n \)
Since \( m^n \) and \( n^m \) must equal \( 2^5 \cdot 5^2 \), we can assume \( m \) and \( n \) are integers that can take values from the prime factorization.

### Step 4: Assign values based on conditions
Given that \( m < n \), we can try assigning \( m = 2 \) and \( n = 5 \):

\[ m = 2 \quad \text{and} \quad n = 5 \]

### Step 5: Verify the values
Now let's verify if these values satisfy the original equation:

\[ m^n = 2^5 \quad \text{and} \quad n^m = 5^2 \]

Calculating:

\[ m^n \cdot n^m = 2^5 \cdot 5^2 = 800 \]

This confirms that our values are correct.

### Step 6: Calculate \( \frac{n}{m} \)
Now we need to find the value of \( \frac{n}{m} \):

\[ \frac{n}{m} = \frac{5}{2} \]

### Final Answer
Thus, the value of \( \frac{n}{m} \) is \( \frac{5}{2} \).

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