Question
Answer
$$\dfrac{8\sqrt{5}+3\sqrt{3}}{2\sqrt{5}-3\sqrt{3}}=-\dfrac{107+30\sqrt{15}}{7}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{8\sqrt{5}+3\sqrt{3}}{2\sqrt{5}-3\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{8\sqrt{5}+3\sqrt{3}}{2\sqrt{5}-3\sqrt{3}}\frac{2\sqrt{5}+3\sqrt{3}}{2\sqrt{5}+3\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{80+24\sqrt{15}+6\sqrt{15}+27}{20+6\sqrt{15}-6\sqrt{15}-27} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{107+30\sqrt{15}}{-7} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{107+30\sqrt{15}}{7}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 2 \sqrt{5} + 3 \sqrt{3}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 8 \sqrt{5} + 3 \sqrt{3}\right) } \cdot \left( 2 \sqrt{5} + 3 \sqrt{3}\right) = \color{blue}{ 8 \sqrt{5}} \cdot 2 \sqrt{5}+\color{blue}{ 8 \sqrt{5}} \cdot 3 \sqrt{3}+\color{blue}{ 3 \sqrt{3}} \cdot 2 \sqrt{5}+\color{blue}{ 3 \sqrt{3}} \cdot 3 \sqrt{3} = \\ = 80 + 24 \sqrt{15} + 6 \sqrt{15} + 27 $$ Simplify denominator. $$ \color{blue}{ \left( 2 \sqrt{5}- 3 \sqrt{3}\right) } \cdot \left( 2 \sqrt{5} + 3 \sqrt{3}\right) = \color{blue}{ 2 \sqrt{5}} \cdot 2 \sqrt{5}+\color{blue}{ 2 \sqrt{5}} \cdot 3 \sqrt{3}\color{blue}{- 3 \sqrt{3}} \cdot 2 \sqrt{5}\color{blue}{- 3 \sqrt{3}} \cdot 3 \sqrt{3} = \\ = 20 + 6 \sqrt{15}- 6 \sqrt{15}-27 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Place a negative sign in front of a fraction. |
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