Find the value of `m` for which `(25)^(m) div 5^(-3) = 5^5`.

Options

A. 2
B. 1
C. 0
D. `-1`.

Answer

To solve the equation \( (25)^m \div 5^{-3} = 5^5 \), we will follow these steps:

### Step 1: Rewrite 25 in terms of base 5
We know that \( 25 = 5^2 \). Therefore, we can rewrite the expression as:
\[ (25)^m = (5^2)^m \]

### Step 2: Apply the power of a power property
Using the property of exponents \( (a^b)^c = a^{bc} \), we can simplify:
\[ (5^2)^m = 5^{2m} \]
Now our equation looks like:
\[ \frac{5^{2m}}{5^{-3}} = 5^5 \]

### Step 3: Simplify the division of powers
Using the property of exponents \( \frac{a^b}{a^c} = a^{b-c} \), we can simplify the left side:
\[ 5^{2m - (-3)} = 5^{2m + 3} \]
So the equation now is:
\[ 5^{2m + 3} = 5^5 \]

### Step 4: Set the exponents equal to each other
Since the bases are the same, we can set the exponents equal to each other:
\[ 2m + 3 = 5 \]

### Step 5: Solve for \( m \)
To isolate \( m \), we first subtract 3 from both sides:
\[ 2m = 5 - 3 \]
\[ 2m = 2 \]
Now, divide both sides by 2:
\[ m = 1 \]

### Final Answer
The value of \( m \) is \( 1 \).

---