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The given expression is \( z = (1 + i)^{10} \), where \( i \) is the imaginary unit, defined as \( i^2 = -1 \).

To solve this, we can follow these steps:

### Step 1: Convert to Polar Form
The complex number \( 1 + i \) can be converted to polar form. For a complex number \( z = a + bi \), its polar form is given by:
\[
z = r \left( \cos \theta + i \sin \theta \right)
\]
where \( r \) is the modulus and \( \theta \) is the argument.

For \( 1 + i \):
- \( r = \sqrt{1^2 + 1^2} = \sqrt{2} \)
- \( \theta = \tan^{-1} \left( \frac{1}{1} \right) = \frac{\pi}{4} \)

So, in polar form:
\[
1 + i = \sqrt{2} \left( \cos \frac{\pi}{4} + i \sin \frac{\pi}{4} \right)
\]

### Step 2: Apply De Moivre's Theorem
De Moivre's Theorem states that for any complex number in polar form \( r(\cos \theta + i \sin \theta) \), raising it to the power of \( n \) results in:
\[
\left( r \left( \cos \theta + i \sin \theta \right) \right)^n = r^n \left( \cos (n\theta) + i \sin (n\theta) \right)
\]

Thus, applying this to \( (1 + i)^{10} \):
\[
(1 + i)^{10} = \left( \sqrt{2} \right)^{10} \left( \cos \left( 10 \times \frac{\pi}{4} \right) + i \sin \left( 10 \times \frac{\pi}{4} \right) \right)
\]

### Step 3: Simplify
- \( \left( \sqrt{2} \right)^{10} = 2^5 = 32 \)
- \( 10 \times \frac{\pi}{4} = \frac{10\pi}{4} = \frac{5\pi}{2} \)

Now, simplify the trigonometric expressions:
- \( \cos \frac{5\pi}{2} = \cos \frac{\pi}{2} = 0 \)
- \( \sin \frac{5\pi}{2} = \sin \frac{\pi}{2} = 1 \)

Thus:
\[
z = 32 (0 + i \cdot 1) = 32i
\]

### Final Answer:
\[
z = 32i
\]

Would you like any further explanation or details? 

Here are some related questions:
1. How do you convert any complex number to polar form?
2. What is the significance of De Moivre's theorem in complex numbers?
3. How can you find the argument of a complex number geometrically?
4. What are other applications of polar forms in mathematics?
5. Can you visualize \( 1 + i \) and \( 32i \) on the complex plane?

**Tip:** When solving powers of complex numbers, converting them into polar form often simplifies the process greatly.

Learn how to solve the complex expression (1 + i)^{10} by converting to polar form and applying De Moivre's theorem. This step-by-step guide explains key concepts in complex numbers and exponential functions, suitable for students in Grades 10-12.

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