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

Answer 1

(i) Here, we will use the identity $\left(x+a\right)\left(x+b\right)=x^2+\left(a+b\right)x+ab$
$$\begin{gathered} 102\times 106 \\ =\left(100+2\right)\left(100+6\right) \\ ={100}^2+\left(2+6\right)100+2\times 6 \\ =10000+800+12 \\ =10812 \end{gathered}$$

(ii) Here, we will use the identity $\left(x+a\right)\left(x+b\right)=x^2+\left(a+b\right)x+ab$
$$\begin{gathered} \mathrm{109\; \times \; 107} \\ =\left(100+9\right)\left(100+7\right) \\ ={100}^2+\left(9+7\right)100+9\times 7 \\ =10000+1600+63 \\ =11663 \end{gathered}$$

(iii) Here, we will use the identity $\left(x+a\right)\left(x+b\right)=x^2+\left(a+b\right)x+ab$
$$\begin{gathered} \mathrm{35\; \times \; 37} \\ =\left(30+5\right)\left(30+7\right) \\ ={30}^2+\left(5+7\right)30+5\times 7 \\ =900+360+35 \\ =1295 \end{gathered}$$

(iv) Here, we will use the identity $\left(x+a\right)\left(x+b\right)=x^2+\left(a+b\right)x+ab$
$$\begin{gathered} \mathrm{53\; \times \; 55} \\ =\left(50+3\right)\left(50+5\right) \\ ={50}^2+\left(3+5\right)50+3\times 5 \\ =2500+400+15 \\ =2915 \end{gathered}$$

(v) Here, we will use the identity $\left(x+a\right)\left(x-b\right)=x^2+\left(a-b\right)x-ab$
$$\begin{gathered} \mathrm{103\; \times \; 96} \\ =\left(100+3\right)\left(100-4\right) \\ ={100}^2+\left(3-4\right)100-3\times 4 \\ =10000-100-12 \\ =9888 \end{gathered}$$

(vi) Here, we will use the identity $\left(x+a\right)\left(x+b\right)=x^2+\left(a+b\right)x+ab$
$$\begin{gathered} \mathrm{34\; \times \; 36} \\ =\left(30+4\right)\left(30+6\right) \\ ={30}^2+\left(4+6\right)30+4\times 6 \\ =900+300+24 \\ =1224 \end{gathered}$$

(vii) Here, we will use the identity $\left(x-a\right)\left(x+b\right)=x^2+\left(b-a\right)x-ab$
$$\begin{gathered} \mathrm{994\; \times \; 1006} \\ =\left(1000-6\right)\times \left(1000+6\right) \\ ={1000}^2+\left(6-6\right)\times 1000-6\times 6 \\ =1000000-36 \\ =999964 \end{gathered}$$