Answer
$$\frac{9\sqrt{5}}{4+\sqrt{7}}=4\sqrt{5}-\sqrt{35}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{9\sqrt{5}}{4+\sqrt{7}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{9\sqrt{5}}{4+\sqrt{7}}\frac{4-\sqrt{7}}{4-\sqrt{7}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{36\sqrt{5}-9\sqrt{35}}{16-4\sqrt{7}+4\sqrt{7}-7} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{36\sqrt{5}-9\sqrt{35}}{9} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{4\sqrt{5}-\sqrt{35}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}4\sqrt{5}-\sqrt{35}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 4- \sqrt{7}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 9 \sqrt{5} } \cdot \left( 4- \sqrt{7}\right) = \color{blue}{ 9 \sqrt{5}} \cdot4+\color{blue}{ 9 \sqrt{5}} \cdot- \sqrt{7} = \\ = 36 \sqrt{5}- 9 \sqrt{35} $$ Simplify denominator. $$ \color{blue}{ \left( 4 + \sqrt{7}\right) } \cdot \left( 4- \sqrt{7}\right) = \color{blue}{4} \cdot4+\color{blue}{4} \cdot- \sqrt{7}+\color{blue}{ \sqrt{7}} \cdot4+\color{blue}{ \sqrt{7}} \cdot- \sqrt{7} = \\ = 16- 4 \sqrt{7} + 4 \sqrt{7}-7 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 9. |
| ⑤ | Remove 1 from denominator. |
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