If `Z=(7+i)/(3+4i)` , then find `Z^14`.

Options

A. `2^7`
B. `(-2)^7`
C. `(2^7) i`
D. `(-2^7) i`

Answer

To find \( Z^{14} \) where \( Z = \frac{7+i}{3+4i} \), we will follow these steps:

### Step 1: Simplify \( Z \)
We start by simplifying \( Z \) by multiplying the numerator and the denominator by the conjugate of the denominator.

\[ Z = \frac{7+i}{3+4i} \cdot \frac{3-4i}{3-4i} = \frac{(7+i)(3-4i)}{(3+4i)(3-4i)} \]

### Step 2: Calculate the Denominator
Now, calculate the denominator:

\[ (3+4i)(3-4i) = 3^2 - (4i)^2 = 9 - 16(-1) = 9 + 16 = 25 \]

### Step 3: Calculate the Numerator
Next, calculate the numerator:

\[ (7+i)(3-4i) = 21 - 28i + 3i - 4i^2 = 21 - 25i + 4 = 25 - 25i \]

### Step 4: Combine Results
Now, combine the results:

\[ Z = \frac{25 - 25i}{25} = 1 - i \]

### Step 5: Convert to Polar Form
Next, we convert \( Z \) to polar form. We can express \( Z = 1 - i \) in polar coordinates.

1. Calculate the modulus \( r \):
\[ r = |Z| = \sqrt{1^2 + (-1)^2} = \sqrt{2} \]

2. Calculate the argument \( \theta \):
\[ \theta = \tan^{-1}\left(\frac{-1}{1}\right) = -\frac{\pi}{4} \]

Thus, we can write \( Z \) in polar form:
\[ Z = \sqrt{2} \left( \cos\left(-\frac{\pi}{4}\right) + i \sin\left(-\frac{\pi}{4}\right) \right) = \sqrt{2} e^{-i\frac{\pi}{4}} \]

### Step 6: Calculate \( Z^{14} \)
Now we can find \( Z^{14} \):
\[ Z^{14} = \left(\sqrt{2} e^{-i\frac{\pi}{4}}\right)^{14} = (\sqrt{2})^{14} \cdot e^{-i\frac{14\pi}{4}} = 2^7 \cdot e^{-i\frac{7\pi}{2}} \]

### Step 7: Simplify the Exponential
Now simplify \( e^{-i\frac{7\pi}{2}} \):
\[ -\frac{7\pi}{2} = -3\pi - \frac{\pi}{2} = -3\pi + 2\pi - \frac{\pi}{2} = -\pi - \frac{\pi}{2} = -\frac{3\pi}{2} \]

Thus,
\[ Z^{14} = 2^7 \cdot e^{-i\frac{3\pi}{2}} = 128 \cdot \left( \cos\left(-\frac{3\pi}{2}\right) + i \sin\left(-\frac{3\pi}{2}\right) \right) \]

### Step 8: Evaluate the Trigonometric Functions
We know:
\[ \cos\left(-\frac{3\pi}{2}\right) = 0, \quad \sin\left(-\frac{3\pi}{2}\right) = -1 \]

Thus,
\[ Z^{14} = 128 \cdot (0 - i) = -128i \]

### Final Answer
\[ Z^{14} = -128i \]

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