Theory:

Identity \(- 4\): (a + b)(a − b)= a2b2
 
Let us first, simplify the identity (a + b)(a − b)= a2b2.
 
Multiply the expression, as shown below.
 
pic 1.png
 
Now, we construct a figure to understand the concept.
 
pic 3.png
 
Then we construct a rectangle using the above information.
 
In the given figure, \(AB = AD = a\).

So, the area of square \(ABCD = a^2\).
 
pic 2.png
 
Also, \(SB = DP = b\). Then the area of the rectangle \(SBCT = ab\).
 
Similarly, the area of the rectangle \(DPRC = ab\). And the area of the square \(TQRC = b^2\).
 
Area of the rectangle \(DPQT = ab − b^2\).
 
Hence, \(\text{the area of the rectangle APQS = The area of square ABCD}\) \(\text{– area of rectangle STCB}\) \(\text{+ area of rectangle DPQT}\).
 
=a2ab+(abb2)=a2ab+abb2=a2b2
 
Therefore, (a + b)(a − b)= a2b2.
Example:
Simplify (3x + 9)(3x  9) using the identity.
 
First, develop the given (3x + 9)(3x  9) expression using the identity (a + b)(a − b)= a2b2.
 
Here, \(a = 3x\); \(b = 9y\).
 
(3x + 9)(3x9)=(3x)2(9)2=32×x2(9)2=9x281
 
Therefore, (3x + 9)(3x  9) \(=\) 9\(x^2 -\)81.