Factorise : `16x^(4)-y^(4)`

Answer

To factorise the expression \( 16x^4 - y^4 \), we can follow these steps:

### Step 1: Recognize the difference of squares
The expression \( 16x^4 - y^4 \) can be recognized as a difference of squares, which has the general form \( a^2 - b^2 = (a + b)(a - b) \).

### Step 2: Rewrite the terms as squares
We can rewrite \( 16x^4 \) and \( y^4 \) as squares:
- \( 16x^4 = (4x^2)^2 \)
- \( y^4 = (y^2)^2 \)

So, we can rewrite the expression as:
\[ (4x^2)^2 - (y^2)^2 \]

### Step 3: Apply the difference of squares formula
Now, we can apply the difference of squares formula:
\[ (4x^2 + y^2)(4x^2 - y^2) \]

### Step 4: Factor the second term further
Next, we notice that \( 4x^2 - y^2 \) is also a difference of squares. We can factor it further:
- \( 4x^2 = (2x)^2 \)
- \( y^2 = (y)^2 \)

So, we can rewrite \( 4x^2 - y^2 \) as:
\[ (2x + y)(2x - y) \]

### Step 5: Combine all factors
Now, we can combine all the factors we have found:
\[ (4x^2 + y^2)(2x + y)(2x - y) \]

Thus, the fully factored form of the expression \( 16x^4 - y^4 \) is:
\[ (4x^2 + y^2)(2x + y)(2x - y) \]

### Final Answer
The factorised form of \( 16x^4 - y^4 \) is:
\[ (4x^2 + y^2)(2x + y)(2x - y) \]