Answer
$$\frac{\sqrt{2}+3\sqrt{5}}{3\sqrt{7}+5\sqrt{3}}=\frac{-3\sqrt{14}+5\sqrt{6}-9\sqrt{35}+15\sqrt{15}}{12}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{2}+3\sqrt{5}}{3\sqrt{7}+5\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{2}+3\sqrt{5}}{3\sqrt{7}+5\sqrt{3}}\frac{3\sqrt{7}-5\sqrt{3}}{3\sqrt{7}-5\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{3\sqrt{14}-5\sqrt{6}+9\sqrt{35}-15\sqrt{15}}{63-15\sqrt{21}+15\sqrt{21}-75} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{3\sqrt{14}-5\sqrt{6}+9\sqrt{35}-15\sqrt{15}}{-12} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-3\sqrt{14}+5\sqrt{6}-9\sqrt{35}+15\sqrt{15}}{12}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 3 \sqrt{7}- 5 \sqrt{3}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{2} + 3 \sqrt{5}\right) } \cdot \left( 3 \sqrt{7}- 5 \sqrt{3}\right) = \color{blue}{ \sqrt{2}} \cdot 3 \sqrt{7}+\color{blue}{ \sqrt{2}} \cdot- 5 \sqrt{3}+\color{blue}{ 3 \sqrt{5}} \cdot 3 \sqrt{7}+\color{blue}{ 3 \sqrt{5}} \cdot- 5 \sqrt{3} = \\ = 3 \sqrt{14}- 5 \sqrt{6} + 9 \sqrt{35}- 15 \sqrt{15} $$ Simplify denominator. $$ \color{blue}{ \left( 3 \sqrt{7} + 5 \sqrt{3}\right) } \cdot \left( 3 \sqrt{7}- 5 \sqrt{3}\right) = \color{blue}{ 3 \sqrt{7}} \cdot 3 \sqrt{7}+\color{blue}{ 3 \sqrt{7}} \cdot- 5 \sqrt{3}+\color{blue}{ 5 \sqrt{3}} \cdot 3 \sqrt{7}+\color{blue}{ 5 \sqrt{3}} \cdot- 5 \sqrt{3} = \\ = 63- 15 \sqrt{21} + 15 \sqrt{21}-75 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
This page was created using
Rationalize Denominator Calculator