Factorise :
`a^(3) -(27/a^(3))`

Answer

To factorise the expression \( a^{3} - \frac{27}{a^{3}} \), we can follow these steps:

### Step 1: Rewrite the expression
The given expression is:
\[ a^{3} - \frac{27}{a^{3}} \]
We can rewrite \( \frac{27}{a^{3}} \) as \( 27 \cdot a^{-3} \). Thus, the expression becomes:
\[ a^{3} - 27 \cdot a^{-3} \]

### Step 2: Recognize the form of the expression
Notice that \( a^{3} - 27 \) can be expressed as a difference of cubes. We can rewrite \( 27 \) as \( 3^{3} \):
\[ a^{3} - 3^{3} \]

### Step 3: 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, let \( x = a \) and \( y = 3 \). Applying the formula:
\[ a^{3} - 3^{3} = (a - 3)(a^{2} + a \cdot 3 + 3^{2}) \]

### Step 4: Simplify the second factor
Now, simplify the second factor:
\[ a^{2} + 3a + 9 \]
Thus, we have:
\[ a^{3} - 27 = (a - 3)(a^{2} + 3a + 9) \]

### Step 5: Combine with the \( a^{-3} \) term
Now, we need to combine this with the \( a^{-3} \) term we factored out earlier:
\[ a^{3} - \frac{27}{a^{3}} = (a - 3)(a^{2} + 3a + 9) \cdot a^{-3} \]
This can be rewritten as:
\[ \frac{(a - 3)(a^{2} + 3a + 9)}{a^{3}} \]

### Final Factorised Form
Thus, the final factorised form of the expression \( a^{3} - \frac{27}{a^{3}} \) is:
\[ \frac{(a - 3)(a^{2} + 3a + 9)}{a^{3}} \]

---