Answer
$$\frac{8}{(\sqrt{6}+2)^2}=\frac{80-32\sqrt{6}}{4}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{8}{(\sqrt{6}+2)^2}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{8}{6+2\sqrt{6}+2\sqrt{6}+4} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{8}{10+4\sqrt{6}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{8}{10+4\sqrt{6}}\frac{10-4\sqrt{6}}{10-4\sqrt{6}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{80-32\sqrt{6}}{100-40\sqrt{6}+40\sqrt{6}-96} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{80-32\sqrt{6}}{4}\end{aligned} $$ | |
| ① | $$ (\sqrt{6}+2)^2 = \left( \sqrt{6} + 2 \right) \cdot \left( \sqrt{6} + 2 \right) = 6 + 2 \sqrt{6} + 2 \sqrt{6} + 4 $$ |
| ② | Simplify numerator and denominator |
| ③ | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 10- 4 \sqrt{6}} $$. |
| ④ | Multiply in a numerator. $$ \color{blue}{ 8 } \cdot \left( 10- 4 \sqrt{6}\right) = \color{blue}{8} \cdot10+\color{blue}{8} \cdot- 4 \sqrt{6} = \\ = 80- 32 \sqrt{6} $$ Simplify denominator. $$ \color{blue}{ \left( 10 + 4 \sqrt{6}\right) } \cdot \left( 10- 4 \sqrt{6}\right) = \color{blue}{10} \cdot10+\color{blue}{10} \cdot- 4 \sqrt{6}+\color{blue}{ 4 \sqrt{6}} \cdot10+\color{blue}{ 4 \sqrt{6}} \cdot- 4 \sqrt{6} = \\ = 100- 40 \sqrt{6} + 40 \sqrt{6}-96 $$ |
| ⑤ | Simplify numerator and denominator |
This page was created using
Rationalize Denominator Calculator