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We are given the complex number in polar form:

\[
z = 32 \left( \cos\left(\frac{\pi}{3}\right) + i \sin\left(\frac{\pi}{3}\right) \right)
\]

To convert this to rectangular form, we use the relationships for converting polar to rectangular coordinates:

\[
z = r(\cos \theta + i \sin \theta)
\]

where \(r\) is the magnitude and \(\theta\) is the argument (angle).

Here, \(r = 32\) and \(\theta = \frac{\pi}{3}\).

First, calculate the cosine and sine of \(\frac{\pi}{3}\):

\[
\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}
\]
\[
\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}
\]

Now, plug these values into the equation for \(z\):

\[
z = 32 \left( \frac{1}{2} + i \frac{\sqrt{3}}{2} \right)
\]

Distribute the 32:

\[
z = 32 \times \frac{1}{2} + i \times 32 \times \frac{\sqrt{3}}{2}
\]
\[
z = 16 + 16i\sqrt{3}
\]

Thus, the rectangular form of the complex number is:

\[
z = 16 + 16i\sqrt{3}
\]

The correct answer from the options provided is:

\[
z = 16 + 16i\sqrt{3}
\]

Let me know if you need more details or have any questions! 

Here are five related questions to expand on this:

1. How do you convert a complex number from rectangular form back to polar form?
2. What are the main differences between polar and rectangular representations of complex numbers?
3. Can you explain how to interpret complex numbers geometrically in the complex plane?
4. What is the general formula for converting from polar to rectangular form?
5. How would the conversion process change if the angle \(\theta\) were negative?

**Tip**: When converting complex numbers, always remember to check the quadrant of the angle to ensure correct signs for cosine and sine!

Use a scientific calculator or unit circle to convert the polar form of the complex number to its equivalent rectangular form.

z = 32 (cos(π/3) + i sin(π/3))

Explore how to convert the complex number 32(cos(π/3) + i sin(π/3)) from polar to rectangular form. Learn key mathematical concepts like trigonometry, polar-to-rectangular conversion, and complex numbers. A step-by-step breakdown helps students in Grades 10-12 grasp the transformation.

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