Sure, let's walk through the steps to solve the problem of expressing [tex]\( z^6 \)[/tex] in rectangular form, where [tex]\( z = 2 \operatorname{cis} 30^\circ \)[/tex].
1. Write [tex]\( z \)[/tex] in exponential form:
We start with the given [tex]\( z = 2 \operatorname{cis} 30^\circ \)[/tex], which in exponential form is written as:
[tex]\[
z = 2 \left( \cos 30^\circ + i \sin 30^\circ \right)
\][/tex]
2. Recall the values of [tex]\(\cos 30^\circ\)[/tex] and [tex]\(\sin 30^\circ\)[/tex]:
[tex]\[
\cos 30^\circ = \frac{\sqrt{3}}{2}, \quad \sin 30^\circ = \frac{1}{2}
\][/tex]
Therefore:
[tex]\[
z = 2 \left( \frac{\sqrt{3}}{2} + i \cdot \frac{1}{2} \right) = \sqrt{3} + i
\][/tex]
3. Express [tex]\( z^6 \)[/tex] using the property of powers in polar form:
When raising a complex number in exponential form to a power, we use:
[tex]\[
z^n = r^n \operatorname{cis} (n \theta)
\][/tex]
Thus for [tex]\( z = 2 \operatorname{cis} 30^\circ \)[/tex]:
[tex]\[
z^6 = (2^6) \operatorname{cis} (6 \times 30^\circ)
\][/tex]
4. Calculate the magnitude and the angle:
[tex]\[
2^6 = 64
\][/tex]
[tex]\[
6 \times 30^\circ = 180^\circ
\][/tex]
So,
[tex]\[
z^6 = 64 \operatorname{cis} 180^\circ
\][/tex]
5. Convert [tex]\( \operatorname{cis} 180^\circ \)[/tex] to rectangular form:
[tex]\[
\operatorname{cis} 180^\circ = \cos 180^\circ + i \sin 180^\circ
\][/tex]
Recall the trigonometric values:
[tex]\[
\cos 180^\circ = -1, \quad \sin 180^\circ = 0
\][/tex]
Therefore:
[tex]\[
\operatorname{cis} 180^\circ = -1
\][/tex]
6. Combine the results:
[tex]\[
z^6 = 64 \cdot (-1) + 64 \cdot i \cdot 0 = -64
\][/tex]
Thus, the rectangular form of [tex]\( z^6 \)[/tex] is [tex]\(-64\)[/tex].
So, the correct option is:
[tex]\[
\boxed{-64}
\][/tex]
