Factorise :
`8a^(3) - 27b^(3)`

Answer

To factorise the expression \(8a^3 - 27b^3\), we can follow these steps:

### Step 1: Identify the cubes
We recognize that \(8a^3\) and \(27b^3\) are both perfect cubes:
- \(8a^3 = (2a)^3\)
- \(27b^3 = (3b)^3\)

### Step 2: Apply the difference of cubes formula
The difference of cubes formula states that:
\[ x^3 - y^3 = (x - y)(x^2 + xy + y^2) \]
In our case, we can let:
- \(x = 2a\)
- \(y = 3b\)

### Step 3: Substitute into the formula
Using the difference of cubes formula, we substitute \(x\) and \(y\):
\[ 8a^3 - 27b^3 = (2a)^3 - (3b)^3 = (2a - 3b)((2a)^2 + (2a)(3b) + (3b)^2) \]

### Step 4: Simplify the expression
Now we simplify the second part of the expression:
- \((2a)^2 = 4a^2\)
- \((2a)(3b) = 6ab\)
- \((3b)^2 = 9b^2\)

Putting it all together, we have:
\[ (2a - 3b)(4a^2 + 6ab + 9b^2) \]

### Final Answer
Thus, the factorised form of \(8a^3 - 27b^3\) is:
\[ (2a - 3b)(4a^2 + 6ab + 9b^2) \]

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