Factorise : `36x^(4) - 84x^(2)y^(2) +49y^(4)`

Answer

To factorise the expression \( 36x^4 - 84x^2y^2 + 49y^4 \), we can follow these steps:

### Step 1: Identify the terms
The expression consists of three terms:
- \( 36x^4 \)
- \( -84x^2y^2 \)
- \( 49y^4 \)

### Step 2: Rewrite the first and last terms as squares
Notice that:
- \( 36x^4 = (6x^2)^2 \)
- \( 49y^4 = (7y^2)^2 \)

So we can rewrite the expression as:
\[ (6x^2)^2 - 84x^2y^2 + (7y^2)^2 \]

### Step 3: Recognize the middle term
The middle term \( -84x^2y^2 \) can be expressed as \( -2 \cdot (6x^2) \cdot (7y^2) \).

### Step 4: Rewrite the expression
Now, we can rewrite the expression as:
\[ (6x^2)^2 - 2 \cdot (6x^2)(7y^2) + (7y^2)^2 \]

### Step 5: Identify the perfect square trinomial
This expression is a perfect square trinomial, which can be factored as:
\[ (a - b)^2 \]
where \( a = 6x^2 \) and \( b = 7y^2 \).

### Step 6: Write the factored form
Thus, we can factor the expression as:
\[ (6x^2 - 7y^2)^2 \]

### Final Answer
The factorised form of \( 36x^4 - 84x^2y^2 + 49y^4 \) is:
\[ (6x^2 - 7y^2)(6x^2 - 7y^2) \]
or simply:
\[ (6x^2 - 7y^2)^2 \]