Question
Answer
$$\dfrac{4\sqrt{3}+5\sqrt{2}}{4\sqrt{5}+3\sqrt{2}}=\dfrac{8\sqrt{15}-6\sqrt{6}+10\sqrt{10}-15}{31}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{4\sqrt{3}+5\sqrt{2}}{4\sqrt{5}+3\sqrt{2}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{4\sqrt{3}+5\sqrt{2}}{4\sqrt{5}+3\sqrt{2}}\frac{4\sqrt{5}-3\sqrt{2}}{4\sqrt{5}-3\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{16\sqrt{15}-12\sqrt{6}+20\sqrt{10}-30}{80-12\sqrt{10}+12\sqrt{10}-18} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{16\sqrt{15}-12\sqrt{6}+20\sqrt{10}-30}{62} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{8\sqrt{15}-6\sqrt{6}+10\sqrt{10}-15}{31}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 4 \sqrt{5}- 3 \sqrt{2}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 4 \sqrt{3} + 5 \sqrt{2}\right) } \cdot \left( 4 \sqrt{5}- 3 \sqrt{2}\right) = \color{blue}{ 4 \sqrt{3}} \cdot 4 \sqrt{5}+\color{blue}{ 4 \sqrt{3}} \cdot- 3 \sqrt{2}+\color{blue}{ 5 \sqrt{2}} \cdot 4 \sqrt{5}+\color{blue}{ 5 \sqrt{2}} \cdot- 3 \sqrt{2} = \\ = 16 \sqrt{15}- 12 \sqrt{6} + 20 \sqrt{10}-30 $$ Simplify denominator. $$ \color{blue}{ \left( 4 \sqrt{5} + 3 \sqrt{2}\right) } \cdot \left( 4 \sqrt{5}- 3 \sqrt{2}\right) = \color{blue}{ 4 \sqrt{5}} \cdot 4 \sqrt{5}+\color{blue}{ 4 \sqrt{5}} \cdot- 3 \sqrt{2}+\color{blue}{ 3 \sqrt{2}} \cdot 4 \sqrt{5}+\color{blue}{ 3 \sqrt{2}} \cdot- 3 \sqrt{2} = \\ = 80- 12 \sqrt{10} + 12 \sqrt{10}-18 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
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