Question

# Evaluate the following: (i) 102 × 106 (ii) 109 × 107 (iii) 35 × 37 (iv) 53 × 55 (v) 103 × 96 (vi) 34 × 36 (vii) 994 × 1006

Solution

## (i) Here, we will use the identity $\left(x+a\right)\left(x+b\right)={x}^{2}+\left(a+b\right)x+ab$ $102×106\phantom{\rule{0ex}{0ex}}=\left(100+2\right)\left(100+6\right)\phantom{\rule{0ex}{0ex}}={100}^{2}+\left(2+6\right)100+2×6\phantom{\rule{0ex}{0ex}}=10000+800+12\phantom{\rule{0ex}{0ex}}=10812$ (ii) Here, we will use the identity $\left(x+a\right)\left(x+b\right)={x}^{2}+\left(a+b\right)x+ab$ $109 × 107\phantom{\rule{0ex}{0ex}}=\left(100+9\right)\left(100+7\right)\phantom{\rule{0ex}{0ex}}={100}^{2}+\left(9+7\right)100+9×7\phantom{\rule{0ex}{0ex}}=10000+1600+63\phantom{\rule{0ex}{0ex}}=11663$ (iii) Here, we will use the identity $\left(x+a\right)\left(x+b\right)={x}^{2}+\left(a+b\right)x+ab$ $35 × 37\phantom{\rule{0ex}{0ex}}=\left(30+5\right)\left(30+7\right)\phantom{\rule{0ex}{0ex}}={30}^{2}+\left(5+7\right)30+5×7\phantom{\rule{0ex}{0ex}}=900+360+35\phantom{\rule{0ex}{0ex}}=1295$ (iv) Here, we will use the identity $\left(x+a\right)\left(x+b\right)={x}^{2}+\left(a+b\right)x+ab$ $53 × 55\phantom{\rule{0ex}{0ex}}=\left(50+3\right)\left(50+5\right)\phantom{\rule{0ex}{0ex}}={50}^{2}+\left(3+5\right)50+3×5\phantom{\rule{0ex}{0ex}}=2500+400+15\phantom{\rule{0ex}{0ex}}=2915$ (v) Here, we will use the identity $\left(x+a\right)\left(x-b\right)={x}^{2}+\left(a-b\right)x-ab$ $103 × 96\phantom{\rule{0ex}{0ex}}=\left(100+3\right)\left(100-4\right)\phantom{\rule{0ex}{0ex}}={100}^{2}+\left(3-4\right)100-3×4\phantom{\rule{0ex}{0ex}}=10000-100-12\phantom{\rule{0ex}{0ex}}=9888$ (vi) Here, we will use the identity $\left(x+a\right)\left(x+b\right)={x}^{2}+\left(a+b\right)x+ab$ $34 × 36\phantom{\rule{0ex}{0ex}}=\left(30+4\right)\left(30+6\right)\phantom{\rule{0ex}{0ex}}={30}^{2}+\left(4+6\right)30+4×6\phantom{\rule{0ex}{0ex}}=900+300+24\phantom{\rule{0ex}{0ex}}=1224$ (vii) Here, we will use the identity $\left(x-a\right)\left(x+b\right)={x}^{2}+\left(b-a\right)x-ab$ $994 × 1006\phantom{\rule{0ex}{0ex}}=\left(1000-6\right)×\left(1000+6\right)\phantom{\rule{0ex}{0ex}}={1000}^{2}+\left(6-6\right)×1000-6×6\phantom{\rule{0ex}{0ex}}=1000000-36\phantom{\rule{0ex}{0ex}}=999964$MathematicsRD Sharma (2019, 2020)All

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