Fractorise:
`63a ^(2)b ^(2) -7`

Answer

To factorize the expression \( 63a^2b^2 - 7 \), we can follow these steps:

### Step 1: Identify the common factor
The first step is to identify the common factor in both terms of the expression. Here, both terms \( 63a^2b^2 \) and \( 7 \) have a common factor of \( 7 \).

### Step 2: Factor out the common factor
We can factor out \( 7 \) from the expression:

\[ 63a^2b^2 - 7 = 7(9a^2b^2 - 1) \]

### Step 3: Recognize the difference of squares
Now, we have \( 9a^2b^2 - 1 \) inside the parentheses. This expression can be recognized as a difference of squares, which follows the formula \( A^2 - B^2 = (A + B)(A - B) \).

Here, we can rewrite \( 9a^2b^2 \) as \( (3ab)^2 \) and \( 1 \) as \( 1^2 \).

### Step 4: Apply the difference of squares formula
Using the difference of squares formula:

\[ 9a^2b^2 - 1 = (3ab)^2 - 1^2 = (3ab + 1)(3ab - 1) \]

### Step 5: Combine the factors
Now, we can combine the factors we found:

\[ 63a^2b^2 - 7 = 7(3ab + 1)(3ab - 1) \]

### Final Answer
Thus, the factorization of \( 63a^2b^2 - 7 \) is:

\[ 7(3ab + 1)(3ab - 1) \]

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