Math Problem Statement
Consider the complex number z=5⋅√2+5⋅√2⋅i
.
Use De Moivre's Theorem to find z4
.
z4=
Solution
Given the complex number , we need to find using De Moivre's Theorem.
Step 1: Convert to Polar Form
A complex number can be represented in polar form as: where:
- is the modulus
- is the argument
For :
The modulus is calculated as:
The argument is:
Thus, the polar form of is:
Step 2: Apply De Moivre's Theorem
De Moivre's Theorem states that: For :
Simplifying:
Since and , we have:
Final Answer:
Would you like further details or explanations?
Here are some related questions you might find useful:
- How can we find the modulus of any complex number?
- What are the geometric interpretations of modulus and argument?
- How does De Moivre's Theorem apply to roots of complex numbers?
- What is the significance of the polar form of a complex number?
- 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.
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Math Problem Analysis
Mathematical Concepts
Complex Numbers
Polar Form
De Moivre's Theorem
Formulas
Polar form of a complex number
De Moivre's Theorem
Theorems
De Moivre's Theorem
Suitable Grade Level
Advanced High School
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