Question
Answer
$$\dfrac{\sqrt{15}+2\sqrt{3}}{15-2\sqrt{3}}=\dfrac{5\sqrt{15}+2\sqrt{5}+10\sqrt{3}+4}{71}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{15}+2\sqrt{3}}{15-2\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{15}+2\sqrt{3}}{15-2\sqrt{3}}\frac{15+2\sqrt{3}}{15+2\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{15\sqrt{15}+6\sqrt{5}+30\sqrt{3}+12}{225+30\sqrt{3}-30\sqrt{3}-12} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{15\sqrt{15}+6\sqrt{5}+30\sqrt{3}+12}{213} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{5\sqrt{15}+2\sqrt{5}+10\sqrt{3}+4}{71}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 15 + 2 \sqrt{3}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{15} + 2 \sqrt{3}\right) } \cdot \left( 15 + 2 \sqrt{3}\right) = \color{blue}{ \sqrt{15}} \cdot15+\color{blue}{ \sqrt{15}} \cdot 2 \sqrt{3}+\color{blue}{ 2 \sqrt{3}} \cdot15+\color{blue}{ 2 \sqrt{3}} \cdot 2 \sqrt{3} = \\ = 15 \sqrt{15} + 6 \sqrt{5} + 30 \sqrt{3} + 12 $$ Simplify denominator. $$ \color{blue}{ \left( 15- 2 \sqrt{3}\right) } \cdot \left( 15 + 2 \sqrt{3}\right) = \color{blue}{15} \cdot15+\color{blue}{15} \cdot 2 \sqrt{3}\color{blue}{- 2 \sqrt{3}} \cdot15\color{blue}{- 2 \sqrt{3}} \cdot 2 \sqrt{3} = \\ = 225 + 30 \sqrt{3}- 30 \sqrt{3}-12 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 3. |
This page was created using
Rationalize Denominator Calculator