Consider the complex number *z*=5⋅√2+5⋅√2⋅i

.

Use De Moivre's Theorem to find *z*4

.

*z*4=

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.

Explore how to use De Moivre's Theorem to calculate z^4 for the complex number z = 5√2 + 5√2i. This detailed solution covers converting the complex number to polar form, applying De Moivre's Theorem, and understanding the modulus and argument concepts. Suitable for advanced high school students and those interested in complex number theory.

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