To compute \((1 + i)^{12}\), we can use De Moivre's Theorem. Here’s a step-by-step process to solve it:

### Step 1: Express \( 1 + i \) in polar form.
1. **Magnitude** of \(1 + i\):  
   \[
   r = |1 + i| = \sqrt{1^2 + 1^2} = \sqrt{2}
   \]

2. **Argument** (angle \(\theta\)):  
   Since \(1 + i\) is in the first quadrant, the argument is:
   \[
   \theta = \tan^{-1}\left(\frac{1}{1}\right) = \frac{\pi}{4}
   \]
   So, \(1 + i\) in polar form is:
   \[
   1 + i = \sqrt{2} \text{cis} \left(\frac{\pi}{4}\right)
   \]
   (where \(\text{cis}(\theta)\) stands for \(\cos(\theta) + i\sin(\theta)\)).

### Step 2: Use De Moivre’s Theorem
De Moivre’s Theorem states:
\[
(r \text{cis} \theta)^n = r^n \text{cis} (n \theta)
\]
So for \((1 + i)^{12}\), we have:
\[
(\sqrt{2} \text{cis} \frac{\pi}{4})^{12} = (\sqrt{2})^{12} \text{cis} \left(12 \cdot \frac{\pi}{4}\right)
\]

### Step 3: Simplify the expression
1. **Magnitude**:
   \[
   (\sqrt{2})^{12} = (2^{1/2})^{12} = 2^6 = 64
   \]

2. **Argument**:
   \[
   12 \cdot \frac{\pi}{4} = 3\pi
   \]
   Since \( \text{cis}(3\pi) = \cos(3\pi) + i\sin(3\pi) = -1 + 0i \), we have:
   \[
   \text{cis}(3\pi) = -1
   \]

### Step 4: Final Answer
Thus,
\[
(1 + i)^{12} = 64 \cdot (-1) = -64
\]

So, \((1 + i)^{12} = -64\).

Would you like more details or have any questions?

### Related Questions:
1. What is De Moivre's Theorem and how does it apply to complex numbers?
2. How do you convert a complex number from rectangular to polar form?
3. What are the properties of powers of complex numbers?
4. How can you compute \((1 - i)^{12}\) using a similar method?
5. What happens when you apply De Moivre's theorem for non-integer powers?

**Tip**: When converting complex numbers to polar form, remember to carefully determine the argument based on the quadrant of the complex number.

Learn how to compute the power of a complex number (1 + i)^12 using De Moivre's Theorem. This step-by-step solution includes converting the complex number to polar form, applying the theorem, and simplifying the result. Suitable for advanced high school and early college students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation