Question
Answer
$$(5+2\sqrt{5})\cdot(5+\sqrt{125})=75+35\sqrt{5}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(5+2\sqrt{5})\cdot(5+\sqrt{125})& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(5+2\sqrt{5})\cdot(5+5\sqrt{5}) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}25+25\sqrt{5}+10\sqrt{5}+50 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}75+35\sqrt{5}\end{aligned} $$ | |
| ① | $$ \sqrt{125} =
\sqrt{ 5 ^2 \cdot 5 } =
\sqrt{ 5 ^2 } \, \sqrt{ 5 } =
5 \sqrt{ 5 }$$ |
| ② | $$ \color{blue}{ \left( 5 + 2 \sqrt{5}\right) } \cdot \left( 5 + 5 \sqrt{5}\right) = \color{blue}{5} \cdot5+\color{blue}{5} \cdot 5 \sqrt{5}+\color{blue}{ 2 \sqrt{5}} \cdot5+\color{blue}{ 2 \sqrt{5}} \cdot 5 \sqrt{5} = \\ = 25 + 25 \sqrt{5} + 10 \sqrt{5} + 50 $$ |
| ③ | Combine like terms |
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