• # question_answer23)                 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) $1.05\times 9.5$.

(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 $=\frac{1}{10}\times (100-0.25)$ $=\frac{1}{10}\,\times 99.75$ $=9.975$