Question

The value of $\sqrt{11-\sqrt{112}}=$

• A. $2+\sqrt{7}$
• B. $2-\sqrt{7}$
• C. $\sqrt{7}-2$
• D. none of these

Answer 1

The correct option is C $\sqrt{7}-2$

$11-\sqrt{112}$

$=11-\sqrt{4\times 28}$

$=11-\sqrt{4}\times \sqrt{28}$

$=11-2\times \sqrt{28}$

$=11-2\times \sqrt{7}\times \sqrt{4}$

$=7+4-2\times \sqrt{7}\times \sqrt{4}$

$=(\sqrt{7}{)}^2+(\sqrt{4}{)}^2-2\times \sqrt{7}\times \sqrt{4}$

$=(\sqrt{7}-\sqrt{4}{)}^2$

Thus, $11-\surd 112=(\surd 7-\surd 4{)}^2$

Hence,

$\sqrt{11-\sqrt{112}}=\sqrt{(\sqrt{7}-\sqrt{4}{)}^2}=\sqrt{7}-\sqrt{4}=\sqrt{7}-2$