2. Simplify the following using identities.
a) (109)2 + (91)
b) (200)2 – (100)
c) (1025)2 – (975)2 d) (10)2 + (2012
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a)(109)2+(91)2
=(100+9)2+(100−9)2
By (a+b)2+(a2−b)2=2[a2+b2]
Therefore (100+9)2+(100−9)2
=2[1002+92]
=2[10000+81]
=2[10081]
=20162
b)(200)2−(10a)(109)2+(91)2
=(100+9)2+(100−9)2
By (a+b)2+(a2−b)2=2[a2+b2]
Therefore (100+9)2+(100−9)2
=2[1002+92]
=2[10000+81]
=2[10081]
=20162
b)(200)2−(100)2
By identity a2−b2=(a+b)(a−b)
∴(200)2−(100)2=(200+100)(200−100)
=300×100
=30000
c)(1025)2−(975)2
By (a+b)2−(a2−b)2=
=4ab
=(1000+25)2−(1000−25)2
=4(1000)(25)
=1000×100
=100000
d)(10)2+(20)2
Now a2+b2=(a+b)2−2ab
∴(10+20)2−2×10×20
⇒(30)2−400
⇒900−400
⇒500
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