Question
Answer
$$(5-5i)^4=-2500$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(5-5i)^4& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}625i^4-2500i^3+3750i^2-2500i+625 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}625+2500i-3750-2500i+625 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}-2500\end{aligned} $$ | |
| ① | $$ (5-5i)^4 = (5-5i)^2 \cdot (5-5i)^2 $$ |
| ② | Find $ \left(5-5i\right)^2 $ using formula. $$ (A - B)^2 = \color{blue}{A^2} - 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ 5 } $ and $ B = \color{red}{ 5i }$. $$ \begin{aligned}\left(5-5i\right)^2 = \color{blue}{5^2} -2 \cdot 5 \cdot 5i + \color{red}{\left( 5i \right)^2} = 25-50i+25i^2\end{aligned} $$ |
| ③ | Multiply each term of $ \left( \color{blue}{25-50i+25i^2}\right) $ by each term in $ \left( 25-50i+25i^2\right) $. $$ \left( \color{blue}{25-50i+25i^2}\right) \cdot \left( 25-50i+25i^2\right) = \\ = 625-1250i+625i^2-1250i+2500i^2-1250i^3+625i^2-1250i^3+625i^4 $$ |
| ④ | Combine like terms: $$ 625 \color{blue}{-1250i} + \color{red}{625i^2} \color{blue}{-1250i} + \color{green}{2500i^2} \color{orange}{-1250i^3} + \color{green}{625i^2} \color{orange}{-1250i^3} +625i^4 = \\ = 625i^4 \color{orange}{-2500i^3} + \color{green}{3750i^2} \color{blue}{-2500i} +625 $$ |
| ⑤ | $$ 625i^4 = 625 \cdot i^2 \cdot i^2 =
625 \cdot ( - 1) \cdot ( - 1) =
625 $$ |
| ⑥ | $$ -2500i^3 = -2500 \cdot \color{blue}{i^2} \cdot i =
-2500 \cdot ( \color{blue}{-1}) \cdot i =
2500 \cdot \, i $$ |
| ⑦ | $$ 3750i^2 = 3750 \cdot (-1) = -3750 $$ |
| ⑧ | Combine like terms: $$ \, \color{blue}{ \cancel{2500i}} \, \, \color{blue}{ -\cancel{2500i}} \, \color{green}{-3750} + \color{orange}{625} + \color{orange}{625} = \color{orange}{-2500} $$ |
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