| (a) \[{{71}^{2}}\] |
| (b) \[{{99}^{2}}\] |
| (c) \[{{102}^{2}}\] |
| (d) \[{{998}^{2}}\] |
| (e) \[{{5.2}^{2}}\] |
| (f) \[297\times 303\] |
| (g) \[78\times 82\] |
| (h) \[{{8.9}^{2}}\] |
Answer:
(a) \[{{71}^{2}}={{\left( 70+1 \right)}^{2}}\] Use the identity, \[{{(a+b)}^{2}}={{a}^{2}}+\text{ }2ab+{{b}^{2}}\] \[={{\left( 70 \right)}^{2}}+\text{ }2\left( 70 \right)\left( 1 \right)+{{\left( 1 \right)}^{2}}\] \[=\text{ }4900+140+1\] = 5041 (b) \[{{99}^{2}}={{\left( 1001 \right)}^{2}}\] Use the identity, \[{{(ab)}^{2}}={{a}^{2}}\text{ }2ab+{{b}^{2}}\] \[={{\left( 100 \right)}^{2}}2\left( 100 \right)\left( 1 \right)+1\] \[=\text{ }10000200+1\] \[=\text{ }10001200\] = 9801 (c) \[{{102}^{2}}={{\left( 100+2 \right)}^{2}}\] Use the identity, \[{{(a+b)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}\] \[=\text{ }{{\left( 100 \right)}^{2}}+2\text{ }\left( 100 \right)\left( 2 \right)+{{\left( 2 \right)}^{2}}\] \[=\text{ }10000+400+4\] = 10404 (d) \[{{998}^{2}}={{\left( 10002 \right)}^{2}}\] Use the identity, \[{{(ab)}^{2}}={{a}^{2}}2ab+{{b}^{2}}\] \[={{\left( 1000 \right)}^{2}}2\left( 1000 \right)\left( 2 \right)+{{\left( 2 \right)}^{2}}\] \[=\text{ }10000002\left( 1000 \right)\left( 2 \right)+{{\left( 2 \right)}^{2}}\] \[=10000004000+4\] = 996004 (e) \[{{5.2}^{2}}={{\left( 5+0.2 \right)}^{2}}\] \[={{\left( 5 \right)}^{2}}+2\left( 5 \right)\left( 0.2 \right)+{{\left( 0.2 \right)}^{2}}\] \[=\text{ }25+2+0.04\] \[=\text{ }27+0.04\] = 27.04 (f) \[297\text{ }\times \text{ }303=\text{ }\left( 300\text{ }-\text{ }3 \right)\text{ }\left( 300\text{ }+\text{ }3 \right)\] Use the identity, \[(ab)(a+b)\text{ }={{a}^{2}}{{b}^{2}}\] \[=\text{ }{{\left( 300 \right)}^{2}}{{\left( 3 \right)}^{2}}\] \[=\text{ }900009\] = 89991 (g) \[78\times 82=\left( 802 \right)\times \left( 80+2 \right)\] Use the identity, \[(ab)(a+b)\text{ }={{a}^{2}}{{b}^{2}}\] \[=\text{ }{{\left( 80 \right)}^{2}}{{\left( 2 \right)}^{2}}\] \[=\text{ }6400\text{ }\text{ }4\] = 6396 (h) \[{{8.9}^{2}}={{\left( 90.1 \right)}^{2}}\] Use the identity, \[{{(ab)}^{2}}={{a}^{2}}\text{ }2ab+{{b}^{2}}\] \[=\text{ }{{\left( 9 \right)}^{2}}\text{ }2\text{ }\left( 9 \right)\left( 0.1 \right)+{{\left( 0.1 \right)}^{2}}\] \[=\text{ }811.8+0.01\] = 79.21
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