Question
Answer
$$\dfrac{4\sqrt{3}-5\sqrt{2}}{4\sqrt{3}+3\sqrt{2}}=\dfrac{39-16\sqrt{6}}{15}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{4\sqrt{3}-5\sqrt{2}}{4\sqrt{3}+3\sqrt{2}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{4\sqrt{3}-5\sqrt{2}}{4\sqrt{3}+3\sqrt{2}}\frac{4\sqrt{3}-3\sqrt{2}}{4\sqrt{3}-3\sqrt{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{48-12\sqrt{6}-20\sqrt{6}+30}{48-12\sqrt{6}+12\sqrt{6}-18} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{78-32\sqrt{6}}{30} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{39-16\sqrt{6}}{15}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 4 \sqrt{3}- 3 \sqrt{2}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 4 \sqrt{3}- 5 \sqrt{2}\right) } \cdot \left( 4 \sqrt{3}- 3 \sqrt{2}\right) = \color{blue}{ 4 \sqrt{3}} \cdot 4 \sqrt{3}+\color{blue}{ 4 \sqrt{3}} \cdot- 3 \sqrt{2}\color{blue}{- 5 \sqrt{2}} \cdot 4 \sqrt{3}\color{blue}{- 5 \sqrt{2}} \cdot- 3 \sqrt{2} = \\ = 48- 12 \sqrt{6}- 20 \sqrt{6} + 30 $$ Simplify denominator. $$ \color{blue}{ \left( 4 \sqrt{3} + 3 \sqrt{2}\right) } \cdot \left( 4 \sqrt{3}- 3 \sqrt{2}\right) = \color{blue}{ 4 \sqrt{3}} \cdot 4 \sqrt{3}+\color{blue}{ 4 \sqrt{3}} \cdot- 3 \sqrt{2}+\color{blue}{ 3 \sqrt{2}} \cdot 4 \sqrt{3}+\color{blue}{ 3 \sqrt{2}} \cdot- 3 \sqrt{2} = \\ = 48- 12 \sqrt{6} + 12 \sqrt{6}-18 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
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