Question
Answer
$$\sqrt{50000}-22\sqrt{125}+3\sqrt{5}(2\sqrt{5}-2)=-16\sqrt{5}+30$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\sqrt{50000}-22\sqrt{125}+3\sqrt{5}(2\sqrt{5}-2)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}100\sqrt{5}-110\sqrt{5}+3\sqrt{5}(2\sqrt{5}-2) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}-10\sqrt{5}+3\sqrt{5}(2\sqrt{5}-2) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-10\sqrt{5}+30-6\sqrt{5} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{-10\sqrt{5}+30-6\sqrt{5}}{1} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}-16\sqrt{5}+30\end{aligned} $$ | |
| ① | $$ \sqrt{50000} =
\sqrt{ 100 ^2 \cdot 5 } =
\sqrt{ 100 ^2 } \, \sqrt{ 5 } =
100 \sqrt{ 5 }$$ |
| ② | $$ - 22 \sqrt{125} =
-22 \sqrt{ 5 ^2 \cdot 5 } =
-22 \sqrt{ 5 ^2 } \, \sqrt{ 5 } =
-22 \cdot 5 \sqrt{ 5 } =
-110 \sqrt{ 5 } $$ |
| ③ | Combine like terms |
| ④ | $$ \color{blue}{ 3 \sqrt{5} } \cdot \left( 2 \sqrt{5}-2\right) = \color{blue}{ 3 \sqrt{5}} \cdot 2 \sqrt{5}+\color{blue}{ 3 \sqrt{5}} \cdot-2 = \\ = 30- 6 \sqrt{5} $$ |
| ⑤ | $$ -10\sqrt{5}+30-6\sqrt{5}
= -10\sqrt{5} \cdot \color{blue}{\frac{ 1 }{ 1}} + 30-6\sqrt{5} \cdot \color{blue}{\frac{ 1 }{ 1}}
= \frac{-10\sqrt{5}+30-6\sqrt{5}}{1} $$ |
| ⑥ | Remove 1 from denominator. |
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