To find \((1 - 2i)^7\), let's use De Moivre's Theorem and the polar form of complex numbers. 

### Step 1: Express the Complex Number in Polar Form
The complex number \( 1 - 2i \) can be expressed in polar form as:
\[
z = r(\cos\theta + i\sin\theta)
\]
where:
- \( r = |z| = \sqrt{1^2 + (-2)^2} = \sqrt{5} \)
- \( \theta = \tan^{-1}\left(\frac{-2}{1}\right) = \tan^{-1}(-2) \)

### Step 2: Calculate the Argument \(\theta\)
The angle \(\theta = \tan^{-1}(-2)\) falls in the fourth quadrant. Thus:
\[
\theta \approx -\tan^{-1}(2)
\]

### Step 3: Apply De Moivre's Theorem
De Moivre’s theorem states:
\[
z^n = r^n \left(\cos(n\theta) + i\sin(n\theta)\right)
\]
Let \( n = 7 \):
\[
(1 - 2i)^7 = (\sqrt{5})^7 \left[\cos(7\theta) + i\sin(7\theta)\right]
\]
Calculating \( r^7 = (\sqrt{5})^7 = 5^{7/2} \).

### Step 4: Calculate \( 7\theta \)
\[
7\theta = 7 \times (-\tan^{-1}(2))
\]
Let \( \phi = -7 \times \tan^{-1}(2) \). Now, substitute back:
\[
(1 - 2i)^7 = 5^{7/2} \left[\cos(\phi) + i\sin(\phi)\right]
\]

### Step 5: Approximate or Express in Rectangular Form
We can either leave the answer in polar form:
\[
(1 - 2i)^7 = 5^{7/2}\left(\cos(\phi) + i\sin(\phi)\right)
\]
or calculate \(\cos(\phi)\) and \(\sin(\phi)\) to express it in rectangular form. Using a calculator for \(\phi\):

1. \(\phi \approx -7 \times \tan^{-1}(2) \approx -8.144\) radians.
2. \(\cos(\phi) \approx -0.304\), \(\sin(\phi) \approx -0.953\).

So:
\[
(1 - 2i)^7 \approx 5^{7/2} \times (-0.304 - 0.953i)
\]
\[
\approx -1696.96 - 5320.57i
\]

### Final Answer
\[
(1 - 2i)^7 \approx -1696.96 - 5320.57i
\]

Would you like a detailed explanation of any step or have any questions?

### Related Questions
1. How do you convert a complex number from rectangular to polar form?
2. What is De Moivre's Theorem and when is it used?
3. How do you calculate the argument of a complex number?
4. Can \((1 - 2i)^7\) be expressed in exact trigonometric form?
5. What is the modulus of a complex number, and why is it important?

### Tip
Always check the quadrant when finding the argument of a complex number to ensure the correct angle is used.

Learn how to solve (1 - 2i)^7 using De Moivre's Theorem and the polar form of complex numbers. This step-by-step guide covers the conversion of a complex number to polar form, applying De Moivre’s theorem, and expressing the final answer in rectangular form. Suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation