Question
Answer
$$\dfrac{4}{\sqrt{10}+2\sqrt{7}}=\dfrac{-4\sqrt{10}+8\sqrt{7}}{18}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{4}{\sqrt{10}+2\sqrt{7}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{4}{\sqrt{10}+2\sqrt{7}}\frac{\sqrt{10}-2\sqrt{7}}{\sqrt{10}-2\sqrt{7}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{4\sqrt{10}-8\sqrt{7}}{10-2\sqrt{70}+2\sqrt{70}-28} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{4\sqrt{10}-8\sqrt{7}}{-18} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-4\sqrt{10}+8\sqrt{7}}{18}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{10}- 2 \sqrt{7}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 4 } \cdot \left( \sqrt{10}- 2 \sqrt{7}\right) = \color{blue}{4} \cdot \sqrt{10}+\color{blue}{4} \cdot- 2 \sqrt{7} = \\ = 4 \sqrt{10}- 8 \sqrt{7} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{10} + 2 \sqrt{7}\right) } \cdot \left( \sqrt{10}- 2 \sqrt{7}\right) = \color{blue}{ \sqrt{10}} \cdot \sqrt{10}+\color{blue}{ \sqrt{10}} \cdot- 2 \sqrt{7}+\color{blue}{ 2 \sqrt{7}} \cdot \sqrt{10}+\color{blue}{ 2 \sqrt{7}} \cdot- 2 \sqrt{7} = \\ = 10- 2 \sqrt{70} + 2 \sqrt{70}-28 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
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