If a and b are positive integers, which of the following is equivalent to `(5a)^(3b)-(5a)^(2b)` ?

Options

A. `5^(b)(a^(3)-a^(2))`
B. `(5a)^(2b)[(5a)^(3b)-1]`
C. `(5a)^(2b)(25a-1)`
D. `(5a)^(2b)[(5a)^(b)-1]`

Answer

To solve the expression \((5a)^{3b} - (5a)^{2b}\), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ (5a)^{3b} - (5a)^{2b} \]

### Step 2: Factor out the common term
Notice that both terms have a common factor of \((5a)^{2b}\). We can factor this out:
\[ (5a)^{2b} \left( (5a)^{3b - 2b} - 1 \right) \]
This simplifies to:
\[ (5a)^{2b} \left( (5a)^{b} - 1 \right) \]

### Step 3: Write the final expression
So, the expression can be rewritten as:
\[ (5a)^{2b} \cdot (5a^{b} - 1) \]

### Step 4: Identify the equivalent expression
Now, we can express the final result in terms of the options provided. The expression \((5a)^{2b}\) can be written as \(5^{2b} \cdot a^{2b}\). Thus, we have:
\[ 5^{2b} \cdot a^{2b} \cdot (5a^{b} - 1) \]

### Conclusion
The equivalent expression to \((5a)^{3b} - (5a)^{2b}\) is:
\[ (5a)^{2b} \cdot (5a^{b} - 1) \]