Factorise :
`48x^(3) - 27x`

Answer

To factorise the expression \(48x^3 - 27x\), we will follow these steps:

### Step 1: Identify the common factor
First, we need to identify the common factor in the terms \(48x^3\) and \(-27x\). Both terms have a common factor of \(3x\).

### Step 2: Factor out the common factor
Now, we will factor out \(3x\) from the expression:
\[ 48x^3 - 27x = 3x(16x^2 - 9) \]

### Step 3: Recognize the difference of squares
Next, we observe that \(16x^2 - 9\) is a difference of squares. We can rewrite it as:
\[ 16x^2 - 9 = (4x)^2 - 3^2 \]

### Step 4: Apply the difference of squares formula
We can apply the difference of squares formula, which states that \(a^2 - b^2 = (a + b)(a - b)\). Here, \(a = 4x\) and \(b = 3\):
\[ (4x)^2 - 3^2 = (4x + 3)(4x - 3) \]

### Step 5: Write the final factorised form
Now, substituting back into our expression, we have:
\[ 48x^3 - 27x = 3x(4x + 3)(4x - 3) \]

Thus, the factorised form of \(48x^3 - 27x\) is:
\[ \boxed{3x(4x + 3)(4x - 3)} \]

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