Answer
$$\frac{7\sqrt{3}-5\sqrt{2}}{\sqrt{56}+\sqrt{12}}=\frac{7\sqrt{42}-21-10\sqrt{7}+5\sqrt{6}}{22}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{7\sqrt{3}-5\sqrt{2}}{\sqrt{56}+\sqrt{12}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{7\sqrt{3}-5\sqrt{2}}{\sqrt{56}+\sqrt{12}}\frac{\sqrt{56}-\sqrt{12}}{\sqrt{56}-\sqrt{12}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{14\sqrt{42}-42-20\sqrt{7}+10\sqrt{6}}{56-4\sqrt{42}+4\sqrt{42}-12} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{14\sqrt{42}-42-20\sqrt{7}+10\sqrt{6}}{44} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{7\sqrt{42}-21-10\sqrt{7}+5\sqrt{6}}{22}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{56}- \sqrt{12}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 7 \sqrt{3}- 5 \sqrt{2}\right) } \cdot \left( \sqrt{56}- \sqrt{12}\right) = \color{blue}{ 7 \sqrt{3}} \cdot \sqrt{56}+\color{blue}{ 7 \sqrt{3}} \cdot- \sqrt{12}\color{blue}{- 5 \sqrt{2}} \cdot \sqrt{56}\color{blue}{- 5 \sqrt{2}} \cdot- \sqrt{12} = \\ = 14 \sqrt{42}-42- 20 \sqrt{7} + 10 \sqrt{6} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{56} + \sqrt{12}\right) } \cdot \left( \sqrt{56}- \sqrt{12}\right) = \color{blue}{ \sqrt{56}} \cdot \sqrt{56}+\color{blue}{ \sqrt{56}} \cdot- \sqrt{12}+\color{blue}{ \sqrt{12}} \cdot \sqrt{56}+\color{blue}{ \sqrt{12}} \cdot- \sqrt{12} = \\ = 56- 4 \sqrt{42} + 4 \sqrt{42}-12 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
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