Factorise :
`a^(3) + 27b^(3)`

Answer

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

### Step 1: Recognize the form of the expression
The expression \( a^3 + 27b^3 \) can be recognized as a sum of cubes. We can rewrite \( 27b^3 \) as \( (3b)^3 \). Thus, we have:
\[ a^3 + (3b)^3 \]

### Step 2: Apply the sum of cubes formula
The sum of cubes can be factored using the identity:
\[ x^3 + y^3 = (x + y)(x^2 - xy + y^2) \]
In our case, let \( x = a \) and \( y = 3b \). Therefore, we can apply the formula:
\[ a^3 + (3b)^3 = (a + 3b)(a^2 - a(3b) + (3b)^2) \]

### Step 3: Simplify the expression
Now we need to simplify the second factor:
1. \( a^2 \) remains as it is.
2. \( -a(3b) = -3ab \).
3. \( (3b)^2 = 9b^2 \).

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

### Step 4: Write the final factored form
Combining everything, the complete factorization of \( a^3 + 27b^3 \) is:
\[ a^3 + 27b^3 = (a + 3b)(a^2 - 3ab + 9b^2) \]

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

---