Solve for m :
` 15^(27) + 15^(2) = 15 ^(2) + 15^(5m+2)`

Answer

To solve the equation \( 15^{27} + 15^{2} = 15^{2} + 15^{5m+2} \), we can follow these steps:

### Step 1: Write the given equation
We start with the equation:
\[ 15^{27} + 15^{2} = 15^{2} + 15^{5m+2} \]

### Step 2: Simplify the equation
We can subtract \( 15^{2} \) from both sides of the equation:
\[ 15^{27} + 15^{2} - 15^{2} = 15^{2} + 15^{5m+2} - 15^{2} \]
This simplifies to:
\[ 15^{27} = 15^{5m+2} \]

### Step 3: Compare the exponents
Since the bases are the same (both are base 15), we can set the exponents equal to each other:
\[ 27 = 5m + 2 \]

### Step 4: Solve for \( m \)
Now, we will isolate \( m \):
1. Subtract 2 from both sides:
\[ 27 - 2 = 5m \]
This gives:
\[ 25 = 5m \]
2. Now, divide both sides by 5:
\[ m = \frac{25}{5} \]
Therefore:
\[ m = 5 \]

### Final Answer
The value of \( m \) is:
\[ \boxed{5} \]