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 (14i)4(\sqrt{14} - i)^4 in both rectangular and exponential forms, we need to follow these steps:

Part 1: Rectangular Form

First, let's compute (14i)4(\sqrt{14} - i)^4 directly in rectangular form. This involves expanding the binomial and simplifying.

Step-by-Step Expansion:

  1. (14i)2=(14i)(14i)(\sqrt{14} - i)^2 = (\sqrt{14} - i)(\sqrt{14} - i)

    Expanding:

    = 14 - 2\sqrt{14}i - 1 = 13 - 2\sqrt{14}i$$
  2. Next, we need to square this result: (13214i)2=(13)2213214i+(214i)2(13 - 2\sqrt{14}i)^2 = (13)^2 - 2 \cdot 13 \cdot 2\sqrt{14}i + (2\sqrt{14}i)^2

    Simplifying each term: (13)2=169(13)^2 = 169 213214i=5214i-2 \cdot 13 \cdot 2\sqrt{14}i = -52\sqrt{14}i (214i)2=414(1)=56(2\sqrt{14}i)^2 = 4 \cdot 14 \cdot (-1) = -56

    So: (13214i)2=1695214i56=1135214i(13 - 2\sqrt{14}i)^2 = 169 - 52\sqrt{14}i - 56 = 113 - 52\sqrt{14}i

Thus, the rectangular form is: 1135214i113 - 52\sqrt{14}i

Part 2: Exponential Form

To express (14i)4(\sqrt{14} - i)^4 in exponential form, we first convert 14i\sqrt{14} - i to its polar form and then raise it to the fourth power.

Step-by-Step Polar Conversion:

  1. Compute the modulus rr: r=(14)2+(1)2=14+1=15r = \sqrt{(\sqrt{14})^2 + (-1)^2} = \sqrt{14 + 1} = \sqrt{15}

  2. Compute the argument θ\theta: θ=tan1(114)\theta = \tan^{-1}\left(\frac{-1}{\sqrt{14}}\right)

    We find: θ=tan1(114)\theta = \tan^{-1}\left(-\frac{1}{\sqrt{14}}\right)

  3. Write the number in polar form: 14i=15eiθ\sqrt{14} - i = \sqrt{15} e^{i\theta}

  4. Raise the polar form to the fourth power: (15eiθ)4=(15)4ei4θ=152ei4θ=225ei4θ(\sqrt{15} e^{i\theta})^4 = (\sqrt{15})^4 e^{i \cdot 4\theta} = 15^2 e^{i \cdot 4\theta} = 225 e^{i \cdot 4\theta}

Given: θ=tan1(114)\theta = \tan^{-1}\left(-\frac{1}{\sqrt{14}}\right)

Finally: 4θ=4tan1(114)4\theta = 4 \tan^{-1}\left(-\frac{1}{\sqrt{14}}\right)

Final Answers

  • Rectangular form: 1135214i113 - 52\sqrt{14}i
  • Exponential form: 225ei4θ225 e^{i \cdot 4\theta}

Do you want any further details or have any questions? Here are some more questions you might find helpful:

  1. How do you convert complex numbers from rectangular to polar form?
  2. What are some common trigonometric identities used in polar conversion?
  3. How do you compute the argument θ\theta for different quadrants?
  4. Can you explain the binomial expansion for complex numbers?
  5. 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