Given the complex number $z = 5\sqrt{2} + 5\sqrt{2}i$, we need to find $z^4$ using De Moivre's Theorem.

Step 1: Convert to Polar Form

A complex number $z = a + bi$ can be represented in polar form as: $z = r(\cos \theta + i \sin \theta)$ where:

• $r = \sqrt{a^2 + b^2}$ is the modulus
• $\theta = \tan^{-1}\left(\frac{b}{a}\right)$ is the argument

For $z = 5\sqrt{2} + 5\sqrt{2}i$:

• $a = 5\sqrt{2}$
• $b = 5\sqrt{2}$

The modulus $r$ is calculated as: $r = \sqrt{(5\sqrt{2})^2 + (5\sqrt{2})^2} = \sqrt{50 + 50} = \sqrt{100} = 10$

The argument $\theta$ is: $\theta = \tan^{-1}\left(\frac{5\sqrt{2}}{5\sqrt{2}}\right) = \tan^{-1}(1) = \frac{\pi}{4}$

Thus, the polar form of $z$ is: $z = 10 \left(\cos \frac{\pi}{4} + i \sin \frac{\pi}{4}\right)$

Step 2: Apply De Moivre's Theorem

De Moivre's Theorem states that: $z^n = r^n \left(\cos(n\theta) + i\sin(n\theta)\right)$ For $z^4$: $z^4 = 10^4 \left(\cos\left(4 \times \frac{\pi}{4}\right) + i \sin\left(4 \times \frac{\pi}{4}\right)\right)$

Simplifying: $z^4 = 10000 \left(\cos \pi + i \sin \pi\right)$

Since $\cos \pi = -1$ and $\sin \pi = 0$, we have: $z^4 = 10000(-1 + 0i) = -10000$

Final Answer:

$z^4 = -10000$

Would you like further details or explanations?

Here are some related questions you might find useful:

1. How can we find the modulus of any complex number?
2. What are the geometric interpretations of modulus and argument?
3. How does De Moivre's Theorem apply to roots of complex numbers?
4. What is the significance of the polar form of a complex number?
5. Can De Moivre's Theorem be used for non-integer powers?

Tip: When dealing with complex numbers, always check the quadrant of the angle to correctly determine the argument.