Let z = z = 8 (cosine (StartFraction pi Over 3 EndFraction) + I sine (StartFraction pi Over 3 EndFraction) ) and w = 3 (cosine (StartFraction pi Over 6 EndFraction) + I sine (StartFraction pi Over 6 EndFraction) ).

Which statement describes the geometric construction of the product zw on the complex plane?

Stretch z by a factor of 3 and rotate StartFraction pi Over 2 EndFraction radians counterclockwise.

Stretch z by a factor of 3 and rotate StartFraction pi Over 6 EndFraction radians counterclockwise.

Stretch z by a factor of 24 and rotate StartFraction pi Over 2 EndFraction radians counterclockwise.

Stretch z by a factor of 24 and rotate StartFraction pi Over 6 EndFraction radians counterclockwise.

To solve this problem, we need to use the properties of multiplying complex numbers in polar form. When two complex numbers are in polar form, their product is obtained by multiplying their magnitudes and adding their angles.

Given:
- \( z = 8 \left( \cos \frac{\pi}{3} + i \sin \frac{\pi}{3} \right) \)
- \( w = 3 \left( \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} \right) \)

Let's find the magnitude and argument of \( z \cdot w \).

### Step 1: Magnitude of \( zw \)
The magnitude of \( z \) is \( |z| = 8 \).
The magnitude of \( w \) is \( |w| = 3 \).

The magnitude of the product \( zw \) is:
\[
|zw| = |z| \cdot |w| = 8 \cdot 3 = 24
\]

### Step 2: Argument of \( zw \)
The argument of \( z \) is \( \arg(z) = \frac{\pi}{3} \).
The argument of \( w \) is \( \arg(w) = \frac{\pi}{6} \).

The argument of the product \( zw \) is:
\[
\arg(zw) = \arg(z) + \arg(w) = \frac{\pi}{3} + \frac{\pi}{6} = \frac{\pi}{2}
\]

### Conclusion
The product \( zw \) has a magnitude of 24 and an argument of \( \frac{\pi}{2} \). Thus, on the complex plane, this transformation is represented by **stretching \( z \) by a factor of 3 and rotating by \( \frac{\pi}{2} \) radians counterclockwise**.

### Answer
The correct answer is:
- **Stretch \( z \) by a factor of 3 and rotate \( \frac{\pi}{2} \) radians counterclockwise.**

Would you like additional details or have any questions?

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Here are five additional related questions:
1. How would the construction change if we rotated \( w \) by \( \frac{\pi}{4} \) radians instead?
2. How can we express \( zw \) in rectangular form?
3. What would happen to the construction if \( |w| \) was doubled?
4. How is the rotation angle affected if the angle of \( z \) changes?
5. Can we interpret the rotation direction (counterclockwise) in terms of the unit circle?

**Tip:** When multiplying complex numbers in polar form, always remember to multiply the magnitudes and add the angles for an efficient solution.

Let z = 8 (cos(pi/3) + i sin(pi/3)) and w = 3 (cos(pi/6) + i sin(pi/6)). Which statement describes the geometric construction of the product zw on the complex plane? Stretch z by a factor of 3 and rotate pi/2 radians counterclockwise. Stretch z by a factor of 3 and rotate pi/6 radians counterclockwise. Stretch z by a factor of 24 and rotate pi/2 radians counterclockwise. Stretch z by a factor of 24 and rotate pi/6 radians counterclockwise.

Understand the geometric construction of the product zw, where z = 8 (cos(pi/3) + i sin(pi/3)) and w = 3 (cos(pi/6) + i sin(pi/6)), using polar coordinates. Learn how to calculate the magnitude and argument of zw to identify transformations on the complex plane. Suitable for advanced high school students studying complex numbers.

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