Question
Answer
$$\dfrac{\sqrt{10}+3}{\sqrt{10}-\sqrt{3}}=\dfrac{10+\sqrt{30}+3\sqrt{10}+3\sqrt{3}}{7}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\sqrt{10}+3}{\sqrt{10}-\sqrt{3}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\sqrt{10}+3}{\sqrt{10}-\sqrt{3}}\frac{\sqrt{10}+\sqrt{3}}{\sqrt{10}+\sqrt{3}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{10+\sqrt{30}+3\sqrt{10}+3\sqrt{3}}{10+\sqrt{30}-\sqrt{30}-3} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{10+\sqrt{30}+3\sqrt{10}+3\sqrt{3}}{7}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{10} + \sqrt{3}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( \sqrt{10} + 3\right) } \cdot \left( \sqrt{10} + \sqrt{3}\right) = \color{blue}{ \sqrt{10}} \cdot \sqrt{10}+\color{blue}{ \sqrt{10}} \cdot \sqrt{3}+\color{blue}{3} \cdot \sqrt{10}+\color{blue}{3} \cdot \sqrt{3} = \\ = 10 + \sqrt{30} + 3 \sqrt{10} + 3 \sqrt{3} $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{10}- \sqrt{3}\right) } \cdot \left( \sqrt{10} + \sqrt{3}\right) = \color{blue}{ \sqrt{10}} \cdot \sqrt{10}+\color{blue}{ \sqrt{10}} \cdot \sqrt{3}\color{blue}{- \sqrt{3}} \cdot \sqrt{10}\color{blue}{- \sqrt{3}} \cdot \sqrt{3} = \\ = 10 + \sqrt{30}- \sqrt{30}-3 $$ |
| ③ | Simplify numerator and denominator |
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