We are asked to express $z^3 w^{-1}$ in standard polar form, given:

$z = \frac{\sqrt{3}}{2} + \frac{1}{2}i \quad \text{and} \quad w = -\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}i.$

Step 1: Convert $z$ and $w$ to polar form.

$z = \frac{\sqrt{3}}{2} + \frac{1}{2}i$:

• The modulus $|z|$ is: $|z| = \sqrt{\left(\frac{\sqrt{3}}{2}\right)^2 + \left(\frac{1}{2}\right)^2} = \sqrt{\frac{3}{4} + \frac{1}{4}} = \sqrt{1} = 1.$
• The argument $\arg(z)$ is: $\theta_z = \tan^{-1}\left(\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}\right) = \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6}.$ So, $z = \text{cis}\left(\frac{\pi}{6}\right)$.

$w = -\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}i$:

• The modulus $|w|$ is: $|w| = \sqrt{\left(-\frac{1}{\sqrt{2}}\right)^2 + \left(\frac{1}{\sqrt{2}}\right)^2} = \sqrt{\frac{1}{2} + \frac{1}{2}} = \sqrt{1} = 1.$
• The argument $\arg(w)$ is: $\theta_w = \tan^{-1}\left(\frac{\frac{1}{\sqrt{2}}}{-\frac{1}{\sqrt{2}}}\right) = \tan^{-1}(-1) = \frac{3\pi}{4}.$ So, $w = \text{cis}\left(\frac{3\pi}{4}\right)$.

Step 2: Calculate $z^3$:

$z^3 = \left[\text{cis}\left(\frac{\pi}{6}\right)\right]^3 = \text{cis}\left(3 \times \frac{\pi}{6}\right) = \text{cis}\left(\frac{\pi}{2}\right).$

Step 3: Calculate $w^{-1}$:

$w^{-1} = \left[\text{cis}\left(\frac{3\pi}{4}\right)\right]^{-1} = \text{cis}\left(-\frac{3\pi}{4}\right).$

Step 4: Multiply $z^3$ and $w^{-1}$:

$z^3 w^{-1} = \text{cis}\left(\frac{\pi}{2} + \left(-\frac{3\pi}{4}\right)\right) = \text{cis}\left(\frac{\pi}{2} - \frac{3\pi}{4}\right) = \text{cis}\left(-\frac{\pi}{4}\right).$

Thus, $z^3 w^{-1} = \cos\left(-\frac{\pi}{4}\right) + i \sin\left(-\frac{\pi}{4}\right)$.

Answer:

The correct option is: $\boxed{\cos\left(-\frac{\pi}{4}\right) + i \sin\left(-\frac{\pi}{4}\right)}.$

Do you want more details or have any other questions? Here are 5 related questions to explore:

1. How do you convert a complex number from rectangular form to polar form?
2. What are the properties of complex numbers under multiplication and division in polar form?
3. How do the modulus and argument change when raising a complex number to a power?
4. How can you interpret the geometric meaning of multiplying two complex numbers in polar form?
5. What is the significance of the angle $\theta$ in the polar form of a complex number?

Tip: When working with complex numbers in polar form, always remember that multiplying corresponds to adding angles and dividing corresponds to subtracting angles.