Question

if a=8+3√7 and b=1/8+3√7. show that a square + b square = 254​

Answer 1

Answer:

$Given: \\ a = 8 + 3 \sqrt{7} \\ \therefore \: {a}^{2} = (8 + 3 \sqrt{7})^{2} \\ = {8}^{2} + {(3 \sqrt{7)} }^{2} + 2 \times 8 \times 3 \sqrt{7} \\ = 64 + 63 + 48 \sqrt{7} \\ {a}^{2} = 127 + 48 \sqrt{7}.....(1) \\ b = \frac{1}{8 + 3 \sqrt{7}} \\ {b}^{2} = \frac{1}{(8 + 3 \sqrt{7})^{2}} \\ = \frac{1}{{8}^{2} + {(3 \sqrt{7)} }^{2} + 2 \times 8 \times 3 \sqrt{7} } \\ = \frac{1}{64 + 63 + 48 \sqrt{7} } \\ {b}^{2} = \frac{1}{127 + 48 \sqrt{7} } \\ = \frac{1}{127 + 48 \sqrt{7} } \times \frac{127 - 48 \sqrt{7}}{127 - 48 \sqrt{7}} \\ = \frac{127 - 48 \sqrt{7} }{(127) ^{2} - (48 \sqrt{7} ) ^{2} } \\ = \frac{127 - 48 \sqrt{7} }{16129 - 16128} \\ = \frac{127 - 48 \sqrt{7} }{1} \\ {b}^{2} = 127 - 48 \sqrt{7} ......(2) \\ adding \: equations \: (1) \: and \: (2) \\ {a}^{2} + {b}^{2} = 127 + 48 \sqrt{7} + 127 - 48 \sqrt{7} \\ = 127 + 127 \\ \therefore \: {a}^{2} + {b}^{2} = 254 \\ thus \: proved.$