To solve this, let's go through the two parts.

We are given: $\left( 3 \left( \cos \frac{\pi}{10} + i \sin \frac{\pi}{10} \right) \right)^5$

This expression is in polar form. To rewrite it in rectangular form $x + yi$ and exponential form $re^{i\theta}$, we need to simplify it step-by-step.

Step 1: Rewrite the Expression Using De Moivre's Theorem

Using De Moivre's theorem, we know that: $\left( r \left( \cos \theta + i \sin \theta \right) \right)^n = r^n \left( \cos(n\theta) + i \sin(n\theta) \right)$ Here:

• $r = 3$
• $\theta = \frac{\pi}{10}$
• $n = 5$

Applying De Moivre's theorem, we get: $\left( 3 \left( \cos \frac{\pi}{10} + i \sin \frac{\pi}{10} \right) \right)^5 = 3^5 \left( \cos \frac{5\pi}{10} + i \sin \frac{5\pi}{10} \right)$

Step 2: Simplify the Magnitude and Angle

Calculate $3^5$: $3^5 = 243$

Now, simplify the angle $\frac{5\pi}{10}$: $\frac{5\pi}{10} = \frac{\pi}{2}$

Thus, our expression becomes: $243 \left( \cos \frac{\pi}{2} + i \sin \frac{\pi}{2} \right)$

Step 3: Substitute and Simplify

Now, substitute $\cos \frac{\pi}{2} = 0$ and $\sin \frac{\pi}{2} = 1$: $243 \left( 0 + i \cdot 1 \right) = 243i$

So, the rectangular form is: $\text{Rectangular form: } 0 + 243i$

Step 4: Write in Exponential Form

The exponential form of a complex number $re^{i\theta}$ is given by the magnitude $r$ and angle $\theta$.

Here:

• The magnitude $r = 243$
• The angle $\theta = \frac{\pi}{2}$

Thus, the exponential form is: $\text{Exponential form: } 243 e^{i \frac{\pi}{2}}$

Final Answer

• Rectangular form: $0 + 243i$
• Exponential form: $243 e^{i \frac{\pi}{2}}$

Would you like further clarification on any step?

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Follow-up Questions

1. How do you determine the angle when using De Moivre's theorem?
2. What is the process for converting a polar form to rectangular form?
3. How do you compute powers of complex numbers in polar form?
4. Why is $\cos \frac{\pi}{2} = 0$ and $\sin \frac{\pi}{2} = 1$?
5. What are the uses of exponential form in complex analysis?

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Tip

When working with complex numbers in polar form, De Moivre's theorem is invaluable for simplifying powers and roots.