Answer
$$\frac{4\sqrt{3}+5\sqrt{2}}{\sqrt{18}+40}=\frac{-6\sqrt{6}+80\sqrt{3}-15+100\sqrt{2}}{791}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{4\sqrt{3}+5\sqrt{2}}{\sqrt{18}+40}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{4\sqrt{3}+5\sqrt{2}}{\sqrt{18}+40}\frac{\sqrt{18}-40}{\sqrt{18}-40} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{12\sqrt{6}-160\sqrt{3}+30-200\sqrt{2}}{18-120\sqrt{2}+120\sqrt{2}-1600} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{12\sqrt{6}-160\sqrt{3}+30-200\sqrt{2}}{-1582} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{6\sqrt{6}-80\sqrt{3}+15-100\sqrt{2}}{-791} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{-6\sqrt{6}+80\sqrt{3}-15+100\sqrt{2}}{791}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{18}-40} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 4 \sqrt{3} + 5 \sqrt{2}\right) } \cdot \left( \sqrt{18}-40\right) = \color{blue}{ 4 \sqrt{3}} \cdot \sqrt{18}+\color{blue}{ 4 \sqrt{3}} \cdot-40+\color{blue}{ 5 \sqrt{2}} \cdot \sqrt{18}+\color{blue}{ 5 \sqrt{2}} \cdot-40 = \\ = 12 \sqrt{6}- 160 \sqrt{3} + 30- 200 \sqrt{2} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{18} + 40\right) } \cdot \left( \sqrt{18}-40\right) = \color{blue}{ \sqrt{18}} \cdot \sqrt{18}+\color{blue}{ \sqrt{18}} \cdot-40+\color{blue}{40} \cdot \sqrt{18}+\color{blue}{40} \cdot-40 = \\ = 18- 120 \sqrt{2} + 120 \sqrt{2}-1600 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
| ⑤ | Multiply both numerator and denominator by -1. |
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