Theory:

Identity –\(3\): (ab)2=a22ab+b2
 
If we replace the variable \((-b)\) in the identity -\((2)\) instead of \((b)\), we get a new identity, that is(ab)2=a22ab+b2.
 
Let us see how we got it.
 
We know the identity –\((2)\), that is (a+b)2=a2+2ab+b2. Substitute the variable \((-b)\) in the identity \((2)\) instead of \((b)\).
 
Then, we get:
 
(a+(b)2=(a+(b))(a+(b))=a2+a(b)+(b)a+(b)2(ab)2=a2+2a(b)+(b)×(b)(ab)2=a22ab+(b)2
 
From the above equation, we can observe that when we change the sign of \(b\), the second term \(2ab\) alone is changed, the other term remains unchanged.
 
Now we see an example to understand this concept better.
Example:
Simplify the expression (3x10y)2 using the appropriate identity.
 
We can simplify the given expression (3x10y)2, using the identity (ab)2=a22ab+b2.
 
Here, \(a =3x\); \(b = 10y\)
 
Now, substitute \(a\), and \(b\) values in the identity.
 
=(3x)22(3x×10y)+(10y)2=9x22(30xy)+100y2=9x260xy+100y2
 
Therefore, (3x10y)2 \(=\) 9x260xy+100y2.