If `m^(n)*n^(m)=5184`, then the value of `m+n` is

Options

A. `5`
B. `7`
C. `4`
D. `8`

Answer

To solve the equation \( m^n \cdot n^m = 5184 \), we will follow these steps:

### Step 1: Prime Factorization of 5184
We start by finding the prime factorization of 5184. We can do this by dividing the number by the smallest prime numbers until we reach 1.

1. Divide by 2:
- \( 5184 \div 2 = 2592 \)
- \( 2592 \div 2 = 1296 \)
- \( 1296 \div 2 = 648 \)
- \( 648 \div 2 = 324 \)
- \( 324 \div 2 = 162 \)
- \( 162 \div 2 = 81 \) (81 is no longer divisible by 2)

2. Now, we divide by 3:
- \( 81 \div 3 = 27 \)
- \( 27 \div 3 = 9 \)
- \( 9 \div 3 = 3 \)
- \( 3 \div 3 = 1 \)

So, the prime factorization of 5184 is:
\[ 5184 = 2^6 \cdot 3^4 \]

### Step 2: Expressing the Equation
Now we rewrite the equation \( m^n \cdot n^m = 5184 \) using the prime factorization we found:
\[ m^n \cdot n^m = 2^6 \cdot 3^4 \]

### Step 3: Assigning Values to m and n
To find suitable values for \( m \) and \( n \), we can assume:
- Let \( m = 4 \) (which is \( 2^2 \))
- Let \( n = 3 \)

Now we check if these values satisfy the equation:
\[ m^n = 4^3 = (2^2)^3 = 2^{2 \cdot 3} = 2^6 \]
\[ n^m = 3^4 \]

Thus,
\[ m^n \cdot n^m = 2^6 \cdot 3^4 \]
This matches with our factorization of 5184.

### Step 4: Finding m + n
Now we calculate \( m + n \):
\[ m + n = 4 + 3 = 7 \]

### Final Answer
The value of \( m + n \) is \( \boxed{7} \).

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