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