7. Using \(a^2 – b^2 = (a + b) (a – b)\), find
(i) \(51^2 – 49^2\)

(ii) \((1.02)^2 – (0.98)^2\)

(iii) \(153^2 – 147^2\)

(iv) \(12.1^2 – 7.9^2\)

(i) \(51^2 – 49^2 = (51 + 49) (51 – 49) \)

\(= 100 \times 2 = 200\)

(ii)\( (1.02)^2 – (0.98)^2 = (1.02 + 0.98) (1.02 – 0.98) \)

\(= 2.00 \times 0.04 = 0.08\)

(iii) \(153^2 – 147^2 = (153 + 147) (153 – 147)\)

\(= 300 \times 6 = 1800\)

(iv) \(12.1^2 – 7.9^2 = (12.1 + 7.9) (12.1 – 7.9)\)

\( = 20.0 \times 4.2 = 84\)

8. Using \((x + a) (x + b) = x^2 + (a + b)x + ab\), find)

(i) \(103 × 104\)

(ii) \(5.1 × 5.2\)

(iii) \(103 × 98\)

(iv) \(9.7 × 9.8\)

(i) \(103 \times 104 = (100 + 3)(100 + 4)\)

\( = (100)^2 + (3 + 4) (100) + 3 \times 4 = 10000 + 700 + 12\)

\( = 10712\)

(ii) \(5.1 \times 5.2 = (5 + 0.1) (5 + 0.2)\)

\( = (5)^2 + (0.1 + 0.2) (5) + 0.1 \times 0.2 \)

\(= 25 + 1.5 + 0.02 = 26.5 + 0.02 = 26.52\)

(iii) \(103 \times 98 = (100 + 3) (100 – 2)\)

\( = (100)^2 + (3 – 2) (100) + 3 \times (-2) = 10000 + 100 – 6 \)

\(= 10100 – 6 = 10094\)

(iv) \(9.7 \times 9.8 = (10 – 0.3) (10 – 0.2) \)

\(= (10)^2 – (0.3 + 0.2) (10) + (-0.3) (-0.2) = 100 – 5 + 0.06 \)

\(= 95 + 0.06 = 95.06\)