Simplify each of the following :

(i) $175 \times 175 + 2 \times 175 \times 25 + 25 \times 25$

(ii) $322 \times 322 - 2 \times 322 \times 22 + 22 \times 22$

(iii) $0.76 \times 0.76 + 2 \times 0.76 \times 0.24 + 0.24 \times 0.24$

(iv) $\frac{7.83 \times 7.83 - 1.17 \times 1.17}{6.66}$

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Solution

$(i) 175 \times 175 + 2 \times 175 \times 25 + 25 \times 25 Let a = 175 and b = 25, then a^{2} + 2 a b + b^{2} = (a + b)^{2} = (175 + 25)^{2} = (200)^{2} = 40000 (i i) 322 \times 322 - 2 \times 322 \times 22 + 22 \times 22 Let a = 322 and b = 22, then a^{2} - 2 a b + b^{2} = (a - b)^{2} = (322 - 22)^{2} = (300)^{2} = 90000 (i i i) 0.76 \times 0.76 + 2 \times 0.76 \times 0.24 + 0.24 \times 0.24 Let a = 0.76 and b = 0.24, then a^{2} + 2 a b + b^{2} = (a + b)^{2} = (0.76 + 0.24)^{2} = (1.00)^{2} = 1 (i v) \frac{7.83 \times 7.83 - 1.17 \times 1.17}{6.66} Let a = 7.83, b = 1.17, a - b = 7.83 - 1.17 = 6.66 \frac{a \times a - b \times b}{a - b} = \frac{a^{2} - b^{2}}{a - b} = \frac{(a + b) (a - b)}{a - b} = a + b = 7.83 + 1.17 = 9.00 = 9$

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