We are tasked with finding the complex cube roots of $3 + 3i$ and expressing the answers in exponential form. To do this, we'll follow these steps:

Step 1: Convert the complex number $3 + 3i$ to polar form.

The polar form of a complex number $z = x + yi$ is written as: $z = r \cdot e^{i\theta}$ where:

• $r = \sqrt{x^2 + y^2}$ is the modulus (magnitude) of the complex number,
• $\theta = \tan^{-1}\left(\frac{y}{x}\right)$ is the argument (angle).

For $3 + 3i$, we calculate:

• $r = \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}$,
• $\theta = \tan^{-1}\left(\frac{3}{3}\right) = \tan^{-1}(1) = \frac{\pi}{4}$.

So, the polar form of $3 + 3i$ is: $3 + 3i = 3\sqrt{2} \cdot e^{i\frac{\pi}{4}}.$

Step 2: Find the cube roots.

The formula for the $n$-th roots of a complex number $z = r \cdot e^{i\theta}$ is: $z_k = \sqrt[n]{r} \cdot e^{i\left(\frac{\theta + 2k\pi}{n}\right)}$ for $k = 0, 1, 2, \dots, n-1$.

For cube roots ($n = 3$), we apply the formula:

• $r = 3\sqrt{2}$, so $\sqrt[3]{r} = \sqrt[3]{3\sqrt{2}} = \left(3\sqrt{2}\right)^{1/3}$,
• The argument $\theta = \frac{\pi}{4}$, and we will calculate the three cube roots for $k = 0, 1, 2$.

Root 1: $k = 0$

$z_0 = \left(3\sqrt{2}\right)^{1/3} \cdot e^{i\frac{\frac{\pi}{4} + 2(0)\pi}{3}} = \left(3\sqrt{2}\right)^{1/3} \cdot e^{i\frac{\pi}{12}}.$

Root 2: $k = 1$

$z_1 = \left(3\sqrt{2}\right)^{1/3} \cdot e^{i\frac{\frac{\pi}{4} + 2(1)\pi}{3}} = \left(3\sqrt{2}\right)^{1/3} \cdot e^{i\frac{7\pi}{12}}.$

Root 3: $k = 2$

$z_2 = \left(3\sqrt{2}\right)^{1/3} \cdot e^{i\frac{\frac{\pi}{4} + 2(2)\pi}{3}} = \left(3\sqrt{2}\right)^{1/3} \cdot e^{i\frac{13\pi}{12}}.$

Final Answer

The complex cube roots of $3 + 3i$ are: $\left(3\sqrt{2}\right)^{1/3} \cdot e^{i\frac{\pi}{12}}, \left(3\sqrt{2}\right)^{1/3} \cdot e^{i\frac{7\pi}{12}}, \left(3\sqrt{2}\right)^{1/3} \cdot e^{i\frac{13\pi}{12}}.$

Would you like to explore more details or have further questions?

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Here are 5 related questions you can try:

1. What are the complex cube roots of $8 + 8i$?
2. Convert the complex number $4 + 4i$ to polar form.
3. How do you find the fourth roots of a complex number?
4. What is the general formula for finding the $n$-th roots of a complex number?
5. Can you explain how the modulus and argument relate to the polar form of a complex number?

Tip:

Always ensure your angle $\theta$ is in the correct range (between 0 and $2\pi$) when writing complex roots in exponential form!