The LCM of `(a^(3)+b^(3))` and `(a^(4)-b^(4))` is :

Options

A. `(a^(3)+b^(3))(a^(2)+b^(2))(a-b)`
B. `(a^(3)+b^(3))(a^(2)+b^(2))(a+b)`
C. `(a^(3)+b^(3))(a^(2)+b^(2)+ab)(a+b)`
D. `(a^(3)+b^(3))(a^(2)-b^(2))(a-b)`

Answer

To find the LCM of \( a^3 + b^3 \) and \( a^4 - b^4 \), we will follow these steps:

### Step 1: Factor \( a^3 + b^3 \)
The expression \( a^3 + b^3 \) can be factored using the sum of cubes formula:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]

### Step 2: Factor \( a^4 - b^4 \)
The expression \( a^4 - b^4 \) can be factored using the difference of squares:
\[ a^4 - b^4 = (a^2 - b^2)(a^2 + b^2) \]
Next, we can factor \( a^2 - b^2 \) further:
\[ a^2 - b^2 = (a - b)(a + b) \]
Thus, we have:
\[ a^4 - b^4 = (a - b)(a + b)(a^2 + b^2) \]

### Step 3: Identify the LCM
Now we need to find the LCM of the two factored forms:
- From \( a^3 + b^3 \): \( (a + b)(a^2 - ab + b^2) \)
- From \( a^4 - b^4 \): \( (a - b)(a + b)(a^2 + b^2) \)

The LCM will include each factor at its highest power:
- The factor \( (a + b) \) appears in both, so we take it once.
- The factor \( (a - b) \) appears only in \( a^4 - b^4 \).
- The factor \( (a^2 - ab + b^2) \) appears only in \( a^3 + b^3 \).
- The factor \( (a^2 + b^2) \) appears only in \( a^4 - b^4 \).

Thus, the LCM is:
\[ \text{LCM} = (a + b)(a - b)(a^2 - ab + b^2)(a^2 + b^2) \]

### Step 4: Simplify the LCM
We can express the LCM in a more compact form:
\[ \text{LCM} = (a + b)(a^2 - ab + b^2)(a - b)(a^2 + b^2) \]
This can also be rearranged as:
\[ \text{LCM} = (a^3 + b^3)(a^2 + b^2)(a - b) \]

### Final Answer
The LCM of \( a^3 + b^3 \) and \( a^4 - b^4 \) is:
\[ \text{LCM} = (a^3 + b^3)(a^2 + b^2)(a - b) \]