Question
Answer
$$\dfrac{15}{5\sqrt{3}+3\sqrt{5}}=\dfrac{5\sqrt{3}-3\sqrt{5}}{2}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{15}{5\sqrt{3}+3\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{15}{5\sqrt{3}+3\sqrt{5}}\frac{5\sqrt{3}-3\sqrt{5}}{5\sqrt{3}-3\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{75\sqrt{3}-45\sqrt{5}}{75-15\sqrt{15}+15\sqrt{15}-45} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{75\sqrt{3}-45\sqrt{5}}{30} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{5\sqrt{3}-3\sqrt{5}}{2}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 5 \sqrt{3}- 3 \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 15 } \cdot \left( 5 \sqrt{3}- 3 \sqrt{5}\right) = \color{blue}{15} \cdot 5 \sqrt{3}+\color{blue}{15} \cdot- 3 \sqrt{5} = \\ = 75 \sqrt{3}- 45 \sqrt{5} $$ Simplify denominator. $$ \color{blue}{ \left( 5 \sqrt{3} + 3 \sqrt{5}\right) } \cdot \left( 5 \sqrt{3}- 3 \sqrt{5}\right) = \color{blue}{ 5 \sqrt{3}} \cdot 5 \sqrt{3}+\color{blue}{ 5 \sqrt{3}} \cdot- 3 \sqrt{5}+\color{blue}{ 3 \sqrt{5}} \cdot 5 \sqrt{3}+\color{blue}{ 3 \sqrt{5}} \cdot- 3 \sqrt{5} = \\ = 75- 15 \sqrt{15} + 15 \sqrt{15}-45 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 15. |
This page was created using
Rationalize Denominator Calculator