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.