Answer
$$(-4+\sqrt{8})\cdot(-4-\sqrt{8})=8$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(-4+\sqrt{8})\cdot(-4-\sqrt{8})& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(-4+2\sqrt{2})\cdot(-4-2\sqrt{2}) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}16+8\sqrt{2}-8\sqrt{2}-8 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}8\end{aligned} $$ | |
| ① | $$ \sqrt{8} =
\sqrt{ 2 ^2 \cdot 2 } =
\sqrt{ 2 ^2 } \, \sqrt{ 2 } =
2 \sqrt{ 2 }$$ |
| ② | $$ - \sqrt{8} =
- \sqrt{ 2 ^2 \cdot 2 } =
- \sqrt{ 2 ^2 } \, \sqrt{ 2 } =
- 2 \sqrt{ 2 }$$ |
| ③ | $$ \color{blue}{ \left( -4 + 2 \sqrt{2}\right) } \cdot \left( -4- 2 \sqrt{2}\right) = \color{blue}{-4} \cdot-4\color{blue}{-4} \cdot- 2 \sqrt{2}+\color{blue}{ 2 \sqrt{2}} \cdot-4+\color{blue}{ 2 \sqrt{2}} \cdot- 2 \sqrt{2} = \\ = 16 + 8 \sqrt{2}- 8 \sqrt{2}-8 $$ |
| ④ | Combine like terms |
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