Question

root 11- root 7 by root 11+ root 7 = a-b root 77 determine the rational numbers a and b

Answer 1

Answer:

$a=\frac{9}{2},\:b=\frac{1}{2}$

Step-by-step explanation:

$LHS=\frac{\sqrt{11}-\sqrt{7}}{\sqrt{11}+\sqrt{7}}$

/* Rationalising the denominator, we get

$=\frac{(\sqrt{11}-\sqrt{7})(\sqrt{11}-\sqrt{7})}{(\sqrt{11}+\sqrt{7})(\sqrt{11}-\sqrt{7})}$

$=\frac{(\sqrt{11}-\sqrt{7})^{2}}{(\sqrt{11})^{2}-(\sqrt{7})^{2}}$

$=\frac{11+7-2\times \sqrt{11}\times \sqrt{7}}{11-7}$

/* By algebraic identities:

i) (a-b)² = a²+b²-2ab

ii) (a+b)(a-b) = a² - b² */

$= \frac{18-2\sqrt{77}}{4}$

$=\frac{2(9-\sqrt{77})}{4}$

$=\frac{9-\sqrt{77}}{2}$

$=\frac{9}{2}-\frac{\sqrt{77}}{2}\\=RHS$

Therefore,

$\frac{9}{2}-\frac{\sqrt{77}}{2}\\=a-b\sqrt{77}$

/* Compare both sides, we get

$a=\frac{9}{2},\:b=\frac{1}{2}$

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Answer 2

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