Question
Answer
$$\dfrac{17\sqrt{3}+5\sqrt{5}}{2\sqrt{3}-5}=-\dfrac{102+85\sqrt{3}+10\sqrt{15}+25\sqrt{5}}{13}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{17\sqrt{3}+5\sqrt{5}}{2\sqrt{3}-5}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{17\sqrt{3}+5\sqrt{5}}{2\sqrt{3}-5}\frac{2\sqrt{3}+5}{2\sqrt{3}+5} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{102+85\sqrt{3}+10\sqrt{15}+25\sqrt{5}}{12+10\sqrt{3}-10\sqrt{3}-25} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{102+85\sqrt{3}+10\sqrt{15}+25\sqrt{5}}{-13} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{102+85\sqrt{3}+10\sqrt{15}+25\sqrt{5}}{13}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 2 \sqrt{3} + 5} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 17 \sqrt{3} + 5 \sqrt{5}\right) } \cdot \left( 2 \sqrt{3} + 5\right) = \color{blue}{ 17 \sqrt{3}} \cdot 2 \sqrt{3}+\color{blue}{ 17 \sqrt{3}} \cdot5+\color{blue}{ 5 \sqrt{5}} \cdot 2 \sqrt{3}+\color{blue}{ 5 \sqrt{5}} \cdot5 = \\ = 102 + 85 \sqrt{3} + 10 \sqrt{15} + 25 \sqrt{5} $$ Simplify denominator. $$ \color{blue}{ \left( 2 \sqrt{3}-5\right) } \cdot \left( 2 \sqrt{3} + 5\right) = \color{blue}{ 2 \sqrt{3}} \cdot 2 \sqrt{3}+\color{blue}{ 2 \sqrt{3}} \cdot5\color{blue}{-5} \cdot 2 \sqrt{3}\color{blue}{-5} \cdot5 = \\ = 12 + 10 \sqrt{3}- 10 \sqrt{3}-25 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Place a negative sign in front of a fraction. |
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