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Find ( a + b ) 4 – ( a – b ) 4 . Hence, evaluate.

Solution :

We space have space open parentheses straight a plus straight b close parentheses to the power of 4 minus open parentheses straight a minus straight b close parentheses to the power of 4 Let space straight a plus straight b equals straight x space and space straight a minus straight b equals straight y rightwards double arrow straight x to the power of 4 minus straight y to the power of 4 equals open parentheses straight x squared close parentheses squared minus open parentheses straight y squared close parentheses squared equals open parentheses straight x squared plus straight y squared close parentheses open parentheses straight x squared minus straight y squared close parentheses space open square brackets As space straight p squared minus straight q squared equals open parentheses straight p plus straight q close parentheses open parentheses straight p minus straight q close parentheses close square brackets equals open parentheses straight x squared plus straight y squared close parentheses open parentheses straight x plus straight y close parentheses open parentheses straight x minus straight y close parentheses equals open curly brackets open parentheses straight a plus straight b close parentheses squared plus open parentheses straight a minus straight b close parentheses squared close curly brackets open parentheses straight a plus straight b plus straight a minus straight b close parentheses open parentheses straight a plus straight b minus straight a plus straight b close parentheses equals open parentheses straight a squared plus straight b squared plus 2 ab plus straight a squared plus straight b squared minus 2 ab close parentheses open parentheses 2 straight a close parentheses open parentheses 2 straight b close parentheses equals open parentheses 2 straight a squared plus 2 straight b squared close parentheses open parentheses 4 ab close parentheses bold rightwards double arrow open parentheses bold a bold plus bold b close parentheses to the power of bold 4 bold minus open parentheses bold a bold minus bold b close parentheses to the power of bold 4 bold equals bold 8 bold ab open parentheses bold a to the power of bold 2 bold plus bold b to the power of bold 2 close parentheses Putting space straight a equals square root of 3 space and space straight b equals square root of 2 comma space we space get comma rightwards double arrow open parentheses square root of 3 plus square root of 2 close parentheses to the power of 4 minus open parentheses square root of 3 minus square root of 2 close parentheses to the power of 4 equals 8 square root of 3. square root of 2 open curly brackets open parentheses square root of 3 close parentheses squared plus open parentheses square root of 2 close parentheses squared close curly brackets equals 8 square root of 6 open parentheses 3 plus 2 close parentheses equals 8 square root of 6 open parentheses 5 close parentheses bold rightwards double arrow open parentheses square root of bold 3 bold plus square root of bold 2 close parentheses to the power of bold 4 bold minus open parentheses square root of bold 3 bold minus square root of bold 2 close parentheses to the power of bold 4 bold equals bold 40 square root of bold 6 bold space open parentheses bold Answer close parentheses

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