Bowers Bowman
05/14/2024 · Senior High School
\( ( \sqrt { 3 \operatorname { cis } } \frac { \pi } { 6 } ) ^ { 10 } \)
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Tutor-Verified Answer
Step-by-step Solution
To simplify the expression \( ( \sqrt{3} \operatorname{cis} \frac{\pi}{6} )^{10} \), we can use the properties of complex numbers in polar form.
First, recall that \( \operatorname{cis} \theta \) is defined as:
\[
\operatorname{cis} \theta = \cos \theta + i \sin \theta
\]
Thus, we can rewrite the expression as:
\[
( \sqrt{3} \operatorname{cis} \frac{\pi}{6} )^{10} = (\sqrt{3})^{10} \cdot \left( \operatorname{cis} \frac{\pi}{6} \right)^{10}
\]
Now, we calculate each part separately.
1. **Calculating \( (\sqrt{3})^{10} \)**:
\[
(\sqrt{3})^{10} = (3^{1/2})^{10} = 3^{10/2} = 3^5 = 243
\]
2. **Calculating \( \left( \operatorname{cis} \frac{\pi}{6} \right)^{10} \)**:
Using the property of exponentiation in polar form, we have:
\[
\left( \operatorname{cis} \frac{\pi}{6} \right)^{10} = \operatorname{cis} \left( 10 \cdot \frac{\pi}{6} \right) = \operatorname{cis} \left( \frac{10\pi}{6} \right) = \operatorname{cis} \left( \frac{5\pi}{3} \right)
\]
3. **Combining the results**:
Now we combine the results from steps 1 and 2:
\[
( \sqrt{3} \operatorname{cis} \frac{\pi}{6} )^{10} = 243 \cdot \operatorname{cis} \left( \frac{5\pi}{3} \right)
\]
Thus, the final result is:
\[
\boxed{243 \operatorname{cis} \frac{5\pi}{3}}
\]
Quick Answer
The simplified form is \( 243 \operatorname{cis} \frac{5\pi}{3} \).
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