Question
Answer
$$\dfrac{-21+4\sqrt{15}}{28+3\sqrt{15}}=\dfrac{-768+175\sqrt{15}}{649}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{-21+4\sqrt{15}}{28+3\sqrt{15}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{-21+4\sqrt{15}}{28+3\sqrt{15}}\frac{28-3\sqrt{15}}{28-3\sqrt{15}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{-588+63\sqrt{15}+112\sqrt{15}-180}{784-84\sqrt{15}+84\sqrt{15}-135} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-768+175\sqrt{15}}{649}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 28- 3 \sqrt{15}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( -21 + 4 \sqrt{15}\right) } \cdot \left( 28- 3 \sqrt{15}\right) = \color{blue}{-21} \cdot28\color{blue}{-21} \cdot- 3 \sqrt{15}+\color{blue}{ 4 \sqrt{15}} \cdot28+\color{blue}{ 4 \sqrt{15}} \cdot- 3 \sqrt{15} = \\ = -588 + 63 \sqrt{15} + 112 \sqrt{15}-180 $$ Simplify denominator. $$ \color{blue}{ \left( 28 + 3 \sqrt{15}\right) } \cdot \left( 28- 3 \sqrt{15}\right) = \color{blue}{28} \cdot28+\color{blue}{28} \cdot- 3 \sqrt{15}+\color{blue}{ 3 \sqrt{15}} \cdot28+\color{blue}{ 3 \sqrt{15}} \cdot- 3 \sqrt{15} = \\ = 784- 84 \sqrt{15} + 84 \sqrt{15}-135 $$ |
| ③ | Simplify numerator and denominator |
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