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🔢 Chapter 4: Exploring Algebraic Identities • Exercise Set 4.3

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Exercise Set 4.1 2Exercise Set 4.2 2Exercise Set 4.3 1Exercise Set 4.4 2Exercise Set 4.5 1End-of-Chapter Exercises 6All Exercises 14
Q1

Exercise Set 4.3, Q1: Find the following squares using one of the above identities. Determine which of these identities will make these calculations easier.
(i) $117^2$
(ii) $78^2$
(iii) $198^2$
(iv) $214^2$
(v) $1104^2$
(vi) $1120^2$

Answer:
(i) $117^2 = (120 - 3)^2 = 14400 - 720 + 9 = \mathbf{13689}$ (Using $(a-b)^2$)
(ii) $78^2 = (80 - 2)^2 = 6400 - 320 + 4 = \mathbf{6084}$ (Using $(a-b)^2$)
(iii) $198^2 = (200 - 2)^2 = 40000 - 800 + 4 = \mathbf{39204}$ (Using $(a-b)^2$)
(iv) $214^2 = (200 + 14)^2 = 40000 + 5600 + 196 = \mathbf{45796}$ (Using $(a+b)^2$)
(v) $1104^2 = (1000 + 100 + 4)^2 = 1000^2 + 100^2 + 4^2 + 2(1000)(100) + 2(1000)(4) + 2(100)(4) = 1000000 + 10000 + 16 + 200000 + 8000 + 800 = \mathbf{1218816}$ (Using $(a+b+c)^2$)
(vi) $1120^2 = (1100 + 20)^2 = 1210000 + 44000 + 400 = \mathbf{1254400}$ (Using $(a+b)^2$)