Find each of the following products
`ab(a^(2) - b^(2))`

Options

A. `a^3b-ab^3`
B. `a^3b+ab^3`
C. `-a^3b+ab^3`
D. `"none"`

Answer

To find the product of the expression \( ab(a^2 - b^2) \), we will follow the distributive property of multiplication. Here’s a step-by-step solution:

### Step 1: Identify the expression
We have the expression \( ab(a^2 - b^2) \).

### Step 2: Apply the distributive property
We will distribute \( ab \) to both terms inside the parentheses \( (a^2 - b^2) \).

\[ ab(a^2 - b^2) = ab \cdot a^2 - ab \cdot b^2 \]

### Step 3: Multiply the first term
Now, we will multiply \( ab \) with \( a^2 \):

\[ ab \cdot a^2 = a^{1}b \cdot a^{2} = a^{1+2}b = a^{3}b \]

### Step 4: Multiply the second term
Next, we will multiply \( ab \) with \( -b^2 \):

\[ -ab \cdot b^2 = -a^{1}b \cdot b^{2} = -a^{1}b^{1+2} = -a^{1}b^{3} \]

### Step 5: Combine the results
Now we combine the results from Step 3 and Step 4:

\[ ab(a^2 - b^2) = a^3b - ab^3 \]

### Final Answer
Thus, the final result of the product \( ab(a^2 - b^2) \) is:

\[ a^3b - ab^3 \]

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