question_answer 23)

                Using identities, evaluate:                 (i) \[{{71}^{2}}\]                                                (ii) \[{{99}^{2}}\]                                               (iii) \[{{102}^{2}}\]                           (iv) \[{{998}^{2}}\]                 (v) \[{{5.2}^{2}}\]                                             (vi) \[297\times 303\]                     (vii) \[78\times 82\]                        (viii) \[{{8.9}^{2}}\]                 (ix) \[1.05\times 9.5\].

Answer:

                (i) \[{{71}^{2}}\]                                                \[{{71}^{2}}={{(70+1)}^{2}}\] \[={{(70)}^{2}}+2(70)\,(1)\,+{{(1)}^{2}}\]                              |Using Identity I \[=4900+140+1\] = 5041 (ii) \[{{99}^{2}}\] \[{{99}^{2}}={{(100-1)}^{2}}\] \[={{(100)}^{2}}-2(100)(1)+{{(1)}^{2}}\] |Using Identity II = 10000 ? 200 + 1 = 9801                 (iii) \[{{102}^{2}}\] \[{{102}^{2}}={{(100+2)}^{2}}\] \[=({{100}^{2}}+2(100)(2)+{{(2)}^{2}}\]  |Using Identity I = 10000 + 400 + 4 = 10404                 (iv) \[{{998}^{2}}\] \[{{998}^{2}}={{(1000-2)}^{2}}\] \[={{(1000)}^{2}}-2(1000)(2)+{{(2)}^{2}}\]             |Using Identity II = 1000000 ? 4000 + 4 = 996004                 (v) \[{{5.2}^{2}}\]                             \[{{5.2}^{2}}={{(5+0.2)}^{2}}\] \[={{(5)}^{2}}+2(5)\,(0.2)\,+{{(0.2)}^{2}}\]            |Using Identity I \[=25+2+0.04\] = 27. 04                 (vi) \[297\times 303\]     \[297\times 303\]\[=(300-3)\times (300+3)\] \[=(300){{}^{2}}-{{(3)}^{2}}\]                                      |Using Identity III = 90000 ? 9 = 89991                 (vii) \[78\times 82\]                                        \[78\times 82\,=(80-2)\times (80+2)\] \[={{(80)}^{2}}-{{(2)}^{2}}\]                                         |Using Identity III = 6400 ? 4 = 6396 (viii) \[{{8.9}^{2}}\] \[{{8.9}^{2}}={{(9-0.1)}^{2}}\] \[={{(9)}^{2}}-2(9)\,(0.1)+{{(0.1)}^{2}}\]                |Using Identity II = 81 ? 1.8 + 0.01 = 79. 21                 (ix) \[1.05\times 9.5\] \[1.05\times 9.5\]\[=\frac{1}{10}\,\times 10.5\times 9.5\] \[=\frac{1}{10}\,(10+0.5)\times (10-0.5)\] \[=\frac{1}{10}\times \{{{(10)}^{2}}-{{(0.5)}^{2}}\}\]                        |Using Identity III \[=\frac{1}{10}\times (100-0.25)\] \[=\frac{1}{10}\,\times 99.75\] \[=9.975\]