Answer
$$\frac{12}{\sqrt{10}-7}=-\frac{12\sqrt{10}+84}{39}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{12}{\sqrt{10}-7}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{12}{\sqrt{10}-7}\frac{\sqrt{10}+7}{\sqrt{10}+7} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{12\sqrt{10}+84}{10+7\sqrt{10}-7\sqrt{10}-49} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{12\sqrt{10}+84}{-39} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-\frac{12\sqrt{10}+84}{39}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ \sqrt{10} + 7} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ 12 } \cdot \left( \sqrt{10} + 7\right) = \color{blue}{12} \cdot \sqrt{10}+\color{blue}{12} \cdot7 = \\ = 12 \sqrt{10} + 84 $$ Simplify denominator. $$ \color{blue}{ \left( \sqrt{10}-7\right) } \cdot \left( \sqrt{10} + 7\right) = \color{blue}{ \sqrt{10}} \cdot \sqrt{10}+\color{blue}{ \sqrt{10}} \cdot7\color{blue}{-7} \cdot \sqrt{10}\color{blue}{-7} \cdot7 = \\ = 10 + 7 \sqrt{10}- 7 \sqrt{10}-49 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Place a negative sign in front of a fraction. |
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