1.Find the square of the following numbers.
(i) 32

(ii) 35

(iii) 86

(iv) 93

(v) 71

(vi) 46


Answer :

(i) We have, \(32 = 30 + 2\)

\((32)^2 = (30 + 2)^2\)

\(= 30(30 + 2) + 2(30 + 2)\)

\(= 30^2 + 30 \times 2 + 2 \times 30 + 2^2\)

\(= 900 + 60 + 60 + 4\)

\(= 1024\)

So we have \((32)^2 = 1024\)

(ii)We have, \(35 = (30 + 5)\)

\((35)^2 = (30 + 5)^2\)

\(= 30(30 + 5) + 5(30 + 5)\)

\(= (30)^2 + 30 \times 5 + 5 \times 30 + (5)^2\)

\(= 900 + 150 + 150 + 25\)

\(= 1225\)

So we have \((35)^2 = 1225\)

(iii) We have, \(86 = (80 + 6)\)

\(86^2 = (80 + 6)^2\)

\(= 80(80 + 6) + 6(80 + 6)\)

\(= (80)^2 + 80 \times 6 + 6 \times 80 + (6)^2\)

\(= 6400 + 480 + 480 + 36\)

\(= 7396\)

So we have, \((86)^2 = 7396\)

(iv)We have, \(93 = (90+ 3)\)

\(93^2 = (90 + 3)^2\)

\(= 90 (90 + 3) + 3(90 + 3)\)

\(= (90)^2 + 90 \times 3 + 3 \times 90 + (3)^2\)

\(= 8100 + 270 + 270 + 9\)

\(= 8649\)

So we have, \((93)^2 = 8649\)

(v) We have, \(71 = (70 + 1)\)

\(71^2 = (70 + 1)^2\)

\(= 70 (70 + 1) + 1(70 + 1)\)

\(= (70)2 + 70 \times 1 + 1 \times 70 + (1)2\)

\(= 4900 + 70 + 70 + 1\)

\(= 5041\)

So we have, \((71)^2 = 5041\)

(vi)We have, \(46 = (40+ 6)\)

\(46^2 = (40 + 6)^2\)

\(= 40 (40 + 6) + 6(40 + 6)\)

\(= (40)^2 + 40 \times 6 + 6 \times 40 + (6)^2\)

\(= 1600 + 240 + 240 + 36\)

\(= 2116\)

So we have, \((46)^2 = 2116\)

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