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Question

# Using identities, evaluate. (i) 712 (ii) 992 (iii) 1022 (iv) 9982 (v) (5.2)2 (vi) 297 × 303 (vii) 78 × 82 (viii) 8.92 (ix) 1.05 × 9.5

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Solution

## (i) 712 = (70 + 1)2 = (70)2 + 2(70) (1) + (1)2 [(a + b)2 = a2 + 2ab + b2 ] = 4900 + 140 + 1 = 5041 (ii) 992 = (100 − 1)2 = (100)2 − 2(100) (1) + (1)2 [(a − b)2 = a2 − 2ab + b2 ] = 10000 − 200 + 1 = 9801 (iii) 1022 = (100 + 2)2 = (100)2 + 2(100)(2) + (2)2 [(a + b)2 = a2 + 2ab + b2 ] = 10000 + 400 + 4 = 10404 (iv) 9982 = (1000 − 2)2 = (1000)2 − 2(1000)(2) + (2)2 [(a − b)2 = a2 − 2ab + b2 ] = 1000000 − 4000 + 4 = 996004 (v) (5.2)2 = (5.0 + 0.2)2 = (5.0)2 + 2(5.0) (0.2) + (0.2)2 [(a + b)2 = a2 + 2ab + b2 ] = 25 + 2 + 0.04 = 27.04 (vi) 297 × 303 = (300 − 3) × (300 + 3) = (300)2 − (3)2 [(a + b) (a − b) = a2 − b2] = 90000 − 9 = 89991 (vii) 78 × 82 = (80 − 2) (80 + 2) = (80)2 − (2)2 [(a + b) (a − b) = a2 − b2] = 6400 − 4 = 6396 (viii) 8.92 = (9.0 − 0.1)2 = (9.0)2 − 2(9.0) (0.1) + (0.1)2 [(a − b)2 = a2 − 2ab + b2 ] = 81 − 1.8 + 0.01 = 79.21 (ix) 1.05 × 9.5 = 1.05 × 0.95 × 10 = (1 + 0.05) (1− 0.05) ×10 = [(1)2 − (0.05)2] × 10 = [1 − 0.0025] × 10 [(a + b) (a − b) = a2 − b2] = 0.9975 × 10 = 9.975

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