Evaluate the following:

Solution:

(i) Here, we will use the identity $(x+a)(x+b)=x^{2}+(a+b) x+a b$

$102 \times 106$

$=(100+2)(100+6)$

$=100^{2}+(2+6) 100+2 \times 6$

$=10000+800+12$

$=10812$

(ii) Here, we will use the identity $(x+a)(x+b)=x^{2}+(a+b) x+a b$

$109 \times 107$

$=(100+9)(100+7)$

$=100^{2}+(9+7) 100+9 \times 7$

$=10000+1600+63$

$=11663$

(iii) Here, we will use the identity $(x+a)(x+b)=x^{2}+(a+b) x+a b$

$35 \times 37$

$=(30+5)(30+7)$

$=30^{2}+(5+7) 30+5 \times 7$

$=900+360+35$

$=1295$

(iv) Here, we will use the identity $(x+a)(x+b)=x^{2}+(a+b) x+a b$

$53 \times 55$

$=(50+3)(50+5)$

$=50^{2}+(3+5) 50+3 \times 5$

$=2500+400+15$

$=2915$

(v) Here, we will use the identity $(x+a)(x-b)=x^{2}+(a-b) x-a b$

$103 \times 96$

$=(100+3)(100-4)$

$=100^{2}+(3-4) 100-3 \times 4$

$=10000-100-12$

$=9888$

(vi) Here, we will use the identity $(x+a)(x+b)=x^{2}+(a+b) x+a b$

$34 \times 36$

$=(30+4)(30+6)$

$=30^{2}+(4+6) 30+4 \times 6$

$=900+300+24$

$=1224$

(vii) Here, we will use the identity $(x-a)(x+b)=x^{2}+(b-a) x-a b$

$994 \times 1006$

$=(1000-6) \times(1000+6)$

$=1000^{2}+(6-6) \times 1000-6 \times 6$

$=1000000-36$

$=999964$