Math Problem Statement
Write the expression in rectangular form, x plus y i , and in exponential form, r e Superscript i theta . left parenthesis StartRoot 14 EndRoot minus i right parenthesis Superscript 4 Question content area bottom Part 1 The rectangular form of the given expression is enter your response here , and the exponential form of the given expression is enter your response here . (Simplify your answers. Use integers or decimals for any numbers in the expressions. Do not round until the final answer. Then round to three decimal places as needed.)
Solution
To express in both rectangular and exponential forms, we need to follow these steps:
Part 1: Rectangular Form
First, let's compute directly in rectangular form. This involves expanding the binomial and simplifying.
Step-by-Step Expansion:
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Expanding:
= 14 - 2\sqrt{14}i - 1 = 13 - 2\sqrt{14}i$$ -
Next, we need to square this result:
Simplifying each term:
So:
Thus, the rectangular form is:
Part 2: Exponential Form
To express in exponential form, we first convert to its polar form and then raise it to the fourth power.
Step-by-Step Polar Conversion:
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Compute the modulus :
-
Compute the argument :
We find:
-
Write the number in polar form:
-
Raise the polar form to the fourth power:
Given:
Finally:
Final Answers
- Rectangular form:
- Exponential form:
Do you want any further details or have any questions? Here are some more questions you might find helpful:
- How do you convert complex numbers from rectangular to polar form?
- What are some common trigonometric identities used in polar conversion?
- How do you compute the argument for different quadrants?
- Can you explain the binomial expansion for complex numbers?
- How do Euler's formula and De Moivre's theorem apply to complex numbers?
Tip: When dealing with complex numbers, always make sure to clearly distinguish between their rectangular and polar forms to avoid confusion.
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Math Problem Analysis
Mathematical Concepts
Complex Numbers
Polar Coordinates
Binomial Expansion
Formulas
(a + bi)^n = r^n (cos(nθ) + i sin(nθ))
r = √(x^2 + y^2)
θ = tan⁻¹(y/x)
Theorems
De Moivre's Theorem
Euler's Formula
Suitable Grade Level
Grades 10-12