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We are asked to use De Moivre's Theorem to express the given complex number in rectangular form.

The complex number provided is:
\[
3\left(\cos\frac{\pi}{6} - i\sin\frac{\pi}{6}\right)^5
\]

### Step 1: Apply De Moivre’s Theorem
De Moivre's Theorem states that for a complex number \( z = r(\cos\theta + i\sin\theta) \), raised to the power \( n \), we have:
\[
z^n = r^n\left(\cos(n\theta) + i\sin(n\theta)\right)
\]

Here, \( r = 3 \), \( \theta = -\frac{\pi}{6} \), and \( n = 5 \).

### Step 2: Calculate the modulus and argument
First, calculate the modulus \( r^5 \):
\[
r^5 = 3^5 = 243
\]

Now, calculate the argument \( 5\theta \):
\[
5\theta = 5 \times \left(-\frac{\pi}{6}\right) = -\frac{5\pi}{6}
\]

### Step 3: Use De Moivre's Theorem
We now have:
\[
z^5 = 243\left(\cos\left(-\frac{5\pi}{6}\right) + i\sin\left(-\frac{5\pi}{6}\right)\right)
\]

### Step 4: Evaluate trigonometric functions
\[
\cos\left(-\frac{5\pi}{6}\right) = -\cos\left(\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2}
\]
\[
\sin\left(-\frac{5\pi}{6}\right) = -\sin\left(\frac{5\pi}{6}\right) = -\frac{1}{2}
\]

### Step 5: Write the expression in rectangular form
Now substitute these values into the expression:
\[
z^5 = 243\left(-\frac{\sqrt{3}}{2} - i\frac{1}{2}\right)
\]

Distribute the 243:
\[
z^5 = 243 \times \left(-\frac{\sqrt{3}}{2}\right) - 243 \times \left(i\frac{1}{2}\right)
\]
\[
z^5 = -\frac{243\sqrt{3}}{2} - i\frac{243}{2}
\]

Thus, the rectangular form of the complex number is:
\[
z^5 = -\frac{243\sqrt{3}}{2} - i\frac{243}{2}
\]

Let me know if you want more details or if you have any questions! Here are some related questions for practice:

1. What is the polar form of \( z = 2 + 2i \)?
2. How does De Moivre's Theorem relate to Euler's formula?
3. What are the real and imaginary parts of \( z^5 = 1 + i \)?
4. How do you convert from rectangular to polar form?
5. What happens to the magnitude of a complex number when raised to a power?

**Tip:** Always simplify trigonometric functions at key angles like \( \frac{\pi}{6} \), \( \frac{\pi}{4} \), etc., as they frequently appear in polar coordinates.

Use de Moivre's Theorem to write each of the following in rectangular form: 3(cos(π/6) - i*sin(π/6))^5

Learn how to apply De Moivre's Theorem to convert the complex number 3(cos(π/6) - i*sin(π/6))^5 into its rectangular form. This problem is ideal for students in Grades 11-12 and covers key concepts in complex numbers and trigonometry.

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