Both of your answers are incorrect.​ First, simplify the given expression using De​ Moivre's Theorem. According to De​ Moivre's Theorem, if

zequals

r e Superscript i theta
 
is a complex​ number, then

z Superscript n

equalsr Superscript n Baseline e Superscript i left parenthesis n theta right parenthesis

. 

To use De​ Moivre's Theorem, first convert the complex number to exponential​ form,

r e Superscript i theta

. 

​Then, apply De​ Moivre's Theorem and simplify.​ Next, to write the given expression in rectangular​ form,

x plus y i

​, 

convert the complex number in exponential form to polar form.​ Then, evaluate the angles and expand. Be sure that

theta

is in the range

0less than or equals

thetaless than2

Let's solve \((\sqrt{14} - i)^4\) using De Moivre's Theorem.

### Step 1: Convert to Exponential Form

To use De Moivre's Theorem, we need the complex number in polar form \(re^{i\theta}\).

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

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

We need to ensure that \(\theta\) is in the correct range \(0 \leq \theta < 2\pi\). Since \(\sqrt{14} - i\) is in the fourth quadrant, the angle will be negative:
\[
\theta = -\tan^{-1}\left(\frac{1}{\sqrt{14}}\right)
\]

#### Write the number in exponential form:
\[
\sqrt{14} - i = \sqrt{15} e^{-i \tan^{-1}\left(\frac{1}{\sqrt{14}}\right)}
\]

### Step 2: Apply De Moivre's Theorem

According to De Moivre's Theorem:
\[
(\sqrt{15} e^{-i \tan^{-1}\left(\frac{1}{\sqrt{14}}\right)})^4 = (\sqrt{15})^4 e^{-i \cdot 4 \tan^{-1}\left(\frac{1}{\sqrt{14}}\right)}
\]

Simplifying the modulus and argument:
\[
(\sqrt{15})^4 = (15)^2 = 225
\]
\[
4 \cdot -\tan^{-1}\left(\frac{1}{\sqrt{14}}\right)
\]

Thus:
\[
(\sqrt{14} - i)^4 = 225 e^{-i \cdot 4 \tan^{-1}\left(\frac{1}{\sqrt{14}}\right)}
\]

### Step 3: Convert to Rectangular Form

To convert the exponential form back to rectangular form, we need to evaluate the trigonometric functions at the angle \( -4 \theta \):

#### Compute \(\cos(4\theta)\) and \(\sin(4\theta)\):
First, find \(\theta\):
\[
\theta = \tan^{-1}\left(\frac{1}{\sqrt{14}}\right) \approx 0.267
\]

Now compute:
\[
4\theta \approx 4 \cdot 0.267 = 1.068
\]

Then:
\[
\cos(4\theta) \approx \cos(1.068) \approx 0.48
\]
\[
\sin(4\theta) \approx \sin(1.068) \approx 0.877
\]

Thus:
\[
225 e^{-i \cdot 4 \theta} = 225 (\cos(4\theta) - i \sin(4\theta)) \approx 225 (0.48 - 0.877i)
\]

### Final Rectangular Form:

\[
225 \cdot 0.48 = 108
\]
\[
225 \cdot (-0.877) = -197.325
\]

Thus:
\[
225 e^{-i \cdot 4 \theta} \approx 108 - 197.325i
\]

### Final Answers:

- Rectangular form: \(108 - 197.325i\)
- Exponential form: \(225 e^{-i \cdot 4 \tan^{-1}\left(\frac{1}{\sqrt{14}}\right)}\)

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

1. How do you derive trigonometric identities for higher multiples of an angle?
2. What is the significance of De Moivre's Theorem in complex analysis?
3. Can you explain the process of converting from polar to rectangular form in more detail?
4. How do you ensure the angle is in the correct range when using inverse trigonometric functions?
5. What are the common pitfalls to avoid when dealing with complex numbers in exponential form?

**Tip:** Always double-check the quadrant of your angle when converting from rectangular to polar form to ensure the correct sign and value of the argument \(\theta\).

Learn how to solve the complex number expression (sqrt(14) - i)^4 using De Moivre's Theorem. This problem covers converting complex numbers to exponential and polar forms, applying De Moivre's Theorem, and converting back to rectangular form. Suitable for advanced high school students and those studying complex analysis.

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