QQuestionMathematics
QuestionMathematics
Find all of the cube roots of 216i and write the answers in rectangular (standard) form.
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Step 1:I'll solve this step-by-step, carefully following the LaTeX formatting guidelines:
Step 2:: Understand the Problem
We need to find the three cube roots of $$216i$$.
This means finding three complex numbers z such that z^{3} = 216i.
Step 3:: Polar Form Conversion
- Argument: $$\arg(216i) = \frac{\pi}{2}
- Magnitude: |216i| = 216
Step 4:: General Cube Root Formula
- $$k = 0, 1, 2
The cube roots will be given by the formula: Where:
Step 5:: Calculate Cube Root Magnitude
\sqrt[3]{216} = 6
Step 6:: Calculate Each Root
= 3\sqrt{3} - 3i
z_{0} = 6\left(\cos\left(\frac{\pi/ 2}{3}\right) + i \sin\left(\frac{\pi/ 2}{3}\right)\right)
Final Answer
1. 3\sqrt{3} + 3i 2. - 3\sqrt{3} - 3i 3. 3\sqrt{3} - 3i
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