Question
Answer
$$\dfrac{3\sqrt{10}-5\sqrt{6}}{4\sqrt{10}+2\sqrt{6}}=\dfrac{45-13\sqrt{15}}{34}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{3\sqrt{10}-5\sqrt{6}}{4\sqrt{10}+2\sqrt{6}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{3\sqrt{10}-5\sqrt{6}}{4\sqrt{10}+2\sqrt{6}}\frac{4\sqrt{10}-2\sqrt{6}}{4\sqrt{10}-2\sqrt{6}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{120-12\sqrt{15}-40\sqrt{15}+60}{160-16\sqrt{15}+16\sqrt{15}-24} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{180-52\sqrt{15}}{136} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{45-13\sqrt{15}}{34}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 4 \sqrt{10}- 2 \sqrt{6}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 3 \sqrt{10}- 5 \sqrt{6}\right) } \cdot \left( 4 \sqrt{10}- 2 \sqrt{6}\right) = \color{blue}{ 3 \sqrt{10}} \cdot 4 \sqrt{10}+\color{blue}{ 3 \sqrt{10}} \cdot- 2 \sqrt{6}\color{blue}{- 5 \sqrt{6}} \cdot 4 \sqrt{10}\color{blue}{- 5 \sqrt{6}} \cdot- 2 \sqrt{6} = \\ = 120- 12 \sqrt{15}- 40 \sqrt{15} + 60 $$ Simplify denominator. $$ \color{blue}{ \left( 4 \sqrt{10} + 2 \sqrt{6}\right) } \cdot \left( 4 \sqrt{10}- 2 \sqrt{6}\right) = \color{blue}{ 4 \sqrt{10}} \cdot 4 \sqrt{10}+\color{blue}{ 4 \sqrt{10}} \cdot- 2 \sqrt{6}+\color{blue}{ 2 \sqrt{6}} \cdot 4 \sqrt{10}+\color{blue}{ 2 \sqrt{6}} \cdot- 2 \sqrt{6} = \\ = 160- 16 \sqrt{15} + 16 \sqrt{15}-24 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 4. |
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