Question
Answer
$$\dfrac{\sqrt{7}+3\sqrt{5}}{\sqrt{7}-3\sqrt{5}}=-\dfrac{26+3\sqrt{35}}{19}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{7}+3\sqrt{5}}{\sqrt{7}-3\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{7}+3\sqrt{5}}{\sqrt{7}-3\sqrt{5}}\frac{\sqrt{7}+3\sqrt{5}}{\sqrt{7}+3\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{7+3\sqrt{35}+3\sqrt{35}+45}{7+3\sqrt{35}-3\sqrt{35}-45} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{52+6\sqrt{35}}{-38} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{26+3\sqrt{35}}{-19} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-\frac{26+3\sqrt{35}}{19}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{7} + 3 \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{7} + 3 \sqrt{5}\right) } \cdot \left( \sqrt{7} + 3 \sqrt{5}\right) = \color{blue}{ \sqrt{7}} \cdot \sqrt{7}+\color{blue}{ \sqrt{7}} \cdot 3 \sqrt{5}+\color{blue}{ 3 \sqrt{5}} \cdot \sqrt{7}+\color{blue}{ 3 \sqrt{5}} \cdot 3 \sqrt{5} = \\ = 7 + 3 \sqrt{35} + 3 \sqrt{35} + 45 $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{7}- 3 \sqrt{5}\right) } \cdot \left( \sqrt{7} + 3 \sqrt{5}\right) = \color{blue}{ \sqrt{7}} \cdot \sqrt{7}+\color{blue}{ \sqrt{7}} \cdot 3 \sqrt{5}\color{blue}{- 3 \sqrt{5}} \cdot \sqrt{7}\color{blue}{- 3 \sqrt{5}} \cdot 3 \sqrt{5} = \\ = 7 + 3 \sqrt{35}- 3 \sqrt{35}-45 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Divide both numerator and denominator by 2. |
| ⑤ | Place a negative sign in front of a fraction. |
This page was created using
Rationalize Denominator Calculator