Answer
$$\frac{5\sqrt{2}+3}{\sqrt{5}-3\sqrt{3}}=-\frac{5\sqrt{10}+15\sqrt{6}+3\sqrt{5}+9\sqrt{3}}{22}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{5\sqrt{2}+3}{\sqrt{5}-3\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{5\sqrt{2}+3}{\sqrt{5}-3\sqrt{3}}\frac{\sqrt{5}+3\sqrt{3}}{\sqrt{5}+3\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{5\sqrt{10}+15\sqrt{6}+3\sqrt{5}+9\sqrt{3}}{5+3\sqrt{15}-3\sqrt{15}-27} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{5\sqrt{10}+15\sqrt{6}+3\sqrt{5}+9\sqrt{3}}{-22} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{5\sqrt{10}+15\sqrt{6}+3\sqrt{5}+9\sqrt{3}}{22}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{5} + 3 \sqrt{3}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 5 \sqrt{2} + 3\right) } \cdot \left( \sqrt{5} + 3 \sqrt{3}\right) = \color{blue}{ 5 \sqrt{2}} \cdot \sqrt{5}+\color{blue}{ 5 \sqrt{2}} \cdot 3 \sqrt{3}+\color{blue}{3} \cdot \sqrt{5}+\color{blue}{3} \cdot 3 \sqrt{3} = \\ = 5 \sqrt{10} + 15 \sqrt{6} + 3 \sqrt{5} + 9 \sqrt{3} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{5}- 3 \sqrt{3}\right) } \cdot \left( \sqrt{5} + 3 \sqrt{3}\right) = \color{blue}{ \sqrt{5}} \cdot \sqrt{5}+\color{blue}{ \sqrt{5}} \cdot 3 \sqrt{3}\color{blue}{- 3 \sqrt{3}} \cdot \sqrt{5}\color{blue}{- 3 \sqrt{3}} \cdot 3 \sqrt{3} = \\ = 5 + 3 \sqrt{15}- 3 \sqrt{15}-27 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Place a negative sign in front of a fraction. |
This page was created using
Rationalize Denominator Calculator