Question
Answer
$$\dfrac{\sqrt{5}}{3\sqrt{3}-3}=\dfrac{\sqrt{15}+\sqrt{5}}{6}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{5}}{3\sqrt{3}-3}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{5}}{3\sqrt{3}-3}\frac{3\sqrt{3}+3}{3\sqrt{3}+3} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{3\sqrt{15}+3\sqrt{5}}{27+9\sqrt{3}-9\sqrt{3}-9} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{3\sqrt{15}+3\sqrt{5}}{18} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{\sqrt{15}+\sqrt{5}}{6}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 3 \sqrt{3} + 3} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \sqrt{5} } \cdot \left( 3 \sqrt{3} + 3\right) = \color{blue}{ \sqrt{5}} \cdot 3 \sqrt{3}+\color{blue}{ \sqrt{5}} \cdot3 = \\ = 3 \sqrt{15} + 3 \sqrt{5} $$ Simplify denominator. $$ \color{blue}{ \left( 3 \sqrt{3}-3\right) } \cdot \left( 3 \sqrt{3} + 3\right) = \color{blue}{ 3 \sqrt{3}} \cdot 3 \sqrt{3}+\color{blue}{ 3 \sqrt{3}} \cdot3\color{blue}{-3} \cdot 3 \sqrt{3}\color{blue}{-3} \cdot3 = \\ = 27 + 9 \sqrt{3}- 9 \sqrt{3}-9 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 3. |
This page was created using
Rationalize Denominator Calculator