Expand the following by using the identity
`(a-b)^(3)=a^(3)-b^(3)-3ab(a-b)`
`(2x-3y)^(3)`

Answer

To expand the expression \((2x - 3y)^{3}\) using the identity \((a - b)^{3} = a^{3} - b^{3} - 3ab(a - b)\), we can follow these steps:

### Step 1: Identify \(a\) and \(b\)
In our case, we can identify:
- \(a = 2x\)
- \(b = 3y\)

### Step 2: Substitute \(a\) and \(b\) into the identity
Now, we substitute \(a\) and \(b\) into the identity:
\[ (2x - 3y)^{3} = (2x)^{3} - (3y)^{3} - 3(2x)(3y)(2x - 3y) \]

### Step 3: Calculate \(a^{3}\) and \(b^{3}\)
Now we calculate \(a^{3}\) and \(b^{3}\):
- \(a^{3} = (2x)^{3} = 8x^{3}\)
- \(b^{3} = (3y)^{3} = 27y^{3}\)

### Step 4: Calculate \(3ab\)
Next, we calculate \(3ab\):
\[ 3ab = 3(2x)(3y) = 18xy \]

### Step 5: Substitute back into the expression
Substituting these values back into the equation, we have:
\[ (2x - 3y)^{3} = 8x^{3} - 27y^{3} - 18xy(2x - 3y) \]

### Step 6: Expand \( -18xy(2x - 3y) \)
Now we need to expand the term \(-18xy(2x - 3y)\):
\[ -18xy(2x) + 18xy(3y) = -36x^{2}y + 54y^{2}x \]

### Step 7: Combine all terms
Putting it all together, we get:
\[ (2x - 3y)^{3} = 8x^{3} - 27y^{3} - 36x^{2}y + 54y^{2}x \]

### Final Answer
Thus, the expanded form of \((2x - 3y)^{3}\) is:
\[ 8x^{3} - 27y^{3} - 36x^{2}y + 54y^{2}x \]

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