Simplify:
`(ax+by)^(3)-(ax-by)^(3)`

Answer

To simplify the expression \((ax + by)^{3} - (ax - by)^{3}\), we can follow these steps:

### Step 1: Apply the Cube Expansion Formula
We will use the identity for the cube of a binomial:
\[ (a + b)^{3} = a^{3} + b^{3} + 3a^{2}b + 3ab^{2} \]
and
\[ (a - b)^{3} = a^{3} - b^{3} - 3a^{2}b + 3ab^{2} \]

### Step 2: Expand Both Cubes
Let \(a = ax\) and \(b = by\). Then we have:
\[ (ax + by)^{3} = (ax)^{3} + (by)^{3} + 3(ax)^{2}(by) + 3(ax)(by)^{2 \]
\[ (ax - by)^{3} = (ax)^{3} - (by)^{3} - 3(ax)^{2}(by) + 3(ax)(by)^{2} \]

### Step 3: Substitute the Expansions into the Original Expression
Now substituting these expansions into the original expression:
\[ (ax + by)^{3} - (ax - by)^{3} = \left[(ax)^{3} + (by)^{3} + 3(ax)^{2}(by) + 3(ax)(by)^{2}\right] - \left[(ax)^{3} - (by)^{3} - 3(ax)^{2}(by) + 3(ax)(by)^{2}\right] \]

### Step 4: Simplify the Expression
Now, let's simplify:
\[ = (ax)^{3} + (by)^{3} + 3(ax)^{2}(by) + 3(ax)(by)^{2} - (ax)^{3} + (by)^{3} + 3(ax)^{2}(by) - 3(ax)(by)^{2} \]
The \((ax)^{3}\) terms cancel out:
\[ = (by)^{3} + (by)^{3} + 3(ax)^{2}(by) + 3(ax)^{2}(by) \]
Combining like terms:
\[ = 2(by)^{3} + 6(ax)^{2}(by) \]

### Step 5: Factor Out Common Terms
Now we can factor out the common terms:
\[ = 2(by)^{3} + 6(ax)^{2}(by) = 2(by)\left[(by)^{2} + 3(ax)^{2}\right] \]

### Final Answer
Thus, the simplified expression is:
\[ 2(by)\left[(by)^{2} + 3(ax)^{2}\right] \]