Question

Arātați cā
$\frac{a + 2b}{2a - b}$
este numār rațional , în fiecare din cazurile:

a)
$a = 11 \sqrt{6} - 6 \sqrt{6} + 8 \sqrt{6} - 7 \sqrt{6 } \\ b = 10 \sqrt{6 } - 12 \sqrt{6} + 3 \sqrt{6} - 5 \sqrt{6}$
b)
$a = 2 \sqrt{2} - 3 \sqrt{2} + 5 \sqrt{2} + \sqrt{2} \\ b = 13 \sqrt{2} - 8 \sqrt{2} - 2 \sqrt{2} + 3 \sqrt{2}$
c)
$a = 11 \sqrt{5} + 4 \sqrt{5} - 9 \sqrt{5} + \sqrt{5} \\ b = 25 \sqrt{5 } - 11 \sqrt{5} - 6 \sqrt{5} - 4 \sqrt{5}$

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Answer 1

$a=11\sqrt{6} -6\sqrt{6} +8\sqrt{6} -7\sqrt{6} \\a=5\sqrt{6} +8\sqrt{6} -7\sqrt{6}\\a=13\sqrt{6} -7\sqrt{6} \\a=6\sqrt{6} \\\\b=10\sqrt{6} -12\sqrt{6} +3\sqrt{6} -5\sqrt{6} \\b=-2\sqrt{6} +3\sqrt{6} -5\sqrt{6} \\b=\sqrt{6} -5\sqrt{6} \\b=-4\sqrt{6} \\\\\frac{a+2b}{2a-b} =\frac{6\sqrt{6}+2*(-4\sqrt{6})}{2*6\sqrt{6}-(-4\sqrt{6} )} =\\\frac{6\sqrt{6}-8\sqrt{6}}{12\sqrt{6}+4\sqrt{6}} =\frac{-2\sqrt{6} }{16\sqrt{6} } =\\-\frac{\sqrt{6} }{8\sqrt{6} } =-\frac{1}{8}$

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$a=2\sqrt{2} -3\sqrt{2} +5\sqrt{2} +\sqrt{2}$

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