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
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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.)

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

### Part 1: Rectangular Form

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

#### Step-by-Step Expansion:

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

   Expanding:
   \[
   (\sqrt{14} - i)^2 = (\sqrt{14})^2 - 2(\sqrt{14})(i) + i^2
   = 14 - 2\sqrt{14}i - 1
   = 13 - 2\sqrt{14}i
   \]

2. Next, we need to square this result:
   \[
   (13 - 2\sqrt{14}i)^2 = (13)^2 - 2 \cdot 13 \cdot 2\sqrt{14}i + (2\sqrt{14}i)^2
   \]

   Simplifying each term:
   \[
   (13)^2 = 169
   \]
   \[
   -2 \cdot 13 \cdot 2\sqrt{14}i = -52\sqrt{14}i
   \]
   \[
   (2\sqrt{14}i)^2 = 4 \cdot 14 \cdot (-1) = -56
   \]

   So:
   \[
   (13 - 2\sqrt{14}i)^2 = 169 - 52\sqrt{14}i - 56 = 113 - 52\sqrt{14}i
   \]

Thus, the rectangular form is:
\[
113 - 52\sqrt{14}i
\]

### Part 2: Exponential Form

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

#### Step-by-Step Polar Conversion:

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

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

   We find:
   \[
   \theta = \tan^{-1}\left(-\frac{1}{\sqrt{14}}\right)
   \]

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

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

Given:
   \[
   \theta = \tan^{-1}\left(-\frac{1}{\sqrt{14}}\right)
   \]

Finally:
   \[
   4\theta = 4 \tan^{-1}\left(-\frac{1}{\sqrt{14}}\right)
   \]

### Final Answers

- Rectangular form: \(113 - 52\sqrt{14}i\)
- Exponential form: \(225 e^{i \cdot 4\theta}\)

#### 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 \(\theta\) 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.

Discover how to convert the complex expression (√14 - i)⁴ into both rectangular form (113 - 52√14i) and exponential form (225 e^(i·4θ)). This tutorial covers key mathematical concepts such as polar coordinates and binomial expansion, perfect for high school students.

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