Question

$4 \div \sqrt{11 - \sqrt{7}$
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Answer 1

Answer:

$\dfrac{4}{\sqrt{11-\sqrt{7}}}=\dfrac{2\left(11+\sqrt{7}\right)\sqrt{11-\sqrt{7}}}{57}\quad \left(\mathrm{Decimal:\quad }\:1.38391\dots \right)$

Step-by-step explanation:

$\dfrac{4}{\sqrt{11-\sqrt{7}}}$

$\gray{\mathrm{Multiply\:by\:the\:conjugate}\:\dfrac{\sqrt{11-\sqrt{7}}}{\sqrt{11-\sqrt{7}}}}$

$=\dfrac{4\sqrt{11-\sqrt{7}}}{\sqrt{11-\sqrt{7}}\sqrt{11-\sqrt{7}}}$

$\black{\sqrt{11-\sqrt{7}}\sqrt{11-\sqrt{7}}=11-\sqrt{7}}$

$=\dfrac{4\sqrt{11-\sqrt{7}}}{11-\sqrt{7}}$

$\gray{\mathrm{Multiply\:by\:the\:conjugate}\:\dfrac{11+\sqrt{7}}{11+\sqrt{7}}}$

$=\dfrac{4\sqrt{11-\sqrt{7}}\left(11+\sqrt{7}\right)}{\left(11-\sqrt{7}\right)\left(11+\sqrt{7}\right)}$

$\black{\left(11-\sqrt{7}\right)\left(11+\sqrt{7}\right)=114}$

$=\dfrac{4\left(11+\sqrt{7}\right)\sqrt{11-\sqrt{7}}}{114}$

$\gray{\mathrm{Cancel\:the\:common\:factor:}\:2}$

$=\dfrac{2\left(11+\sqrt{7}\right)\sqrt{11-\sqrt{7}}}{57}$