Find the squares of the following numbers :
(i) 127 (ii) 503
(iii) 451 (iv) 862
(v) 265

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Solution

$(i) (127)^{2} = (120 + 7)^{2} {(a + b)^{2} = a^{2} + 2 a b + b^{2}} = (120)^{2} + 2 \times 120 \times 7 + (7)^{2} = 14400 + 1680 + 49 = 16129$

$(i i) (503)^{2} = (500 + 3)^{2} {(a + b)^{2} = a^{2} + 2 a b + b^{2}} = (500)^{2} + 2 \times 500 \times 3 + (3)^{2} = 250000 + 3000 + 9 = 253009$

$(i i i) (451)^{2} = (400 + 51)^{2} {(a + b)^{2} = a^{2} + 2 a b + b^{2}} = (400)^{2} + 2 \times 400 \times 51 + (51)^{2} = 160000 + 40800 + 2601 = 203401$

$(i v) (862)^{2} = (800 + 62)^{2} {(a + b)^{2} = a^{2} + 2 a b + b^{2}} = (800)^{2} + 2 \times 800 \times 62 + (62)^{2} = 640000 + 99200 + 3844 = 743044$

$(v) (265)^{2} {(a + b)^{2} = a^{2} + 2 a b + b^{2}} = (200 + 65)^{2} = (200)^{2} + 2 \times 200 \times 65 + (65)^{2} = 40000 + 26000 + 4225 = 70225$

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