Change the following complex numbers into polar form
`(1+ 7i)/((2-i)^(2))`

Answer

To convert the complex number \( z = \frac{1 + 7i}{(2 - i)^2} \) into polar form, we will follow these steps:

### Step 1: Simplify the denominator
First, we need to simplify the denominator \( (2 - i)^2 \).

\[ (2 - i)^2 = 2^2 - 2 \cdot 2 \cdot i + i^2 = 4 - 4i + (-1) = 3 - 4i \]

### Step 2: Rewrite the complex number
Now we can rewrite \( z \):

\[ z = \frac{1 + 7i}{3 - 4i} \]

### Step 3: Multiply numerator and denominator by the conjugate of the denominator
To simplify this further, we multiply the numerator and denominator by the conjugate of the denominator, \( 3 + 4i \):

\[ z = \frac{(1 + 7i)(3 + 4i)}{(3 - 4i)(3 + 4i)} \]

### Step 4: Simplify the denominator using the difference of squares
The denominator simplifies as follows:

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

### Step 5: Expand the numerator
Now, we expand the numerator:

\[ (1 + 7i)(3 + 4i) = 1 \cdot 3 + 1 \cdot 4i + 7i \cdot 3 + 7i \cdot 4i = 3 + 4i + 21i + 28i^2 \]
Since \( i^2 = -1 \), we have:

\[ 28i^2 = 28(-1) = -28 \]
Thus, the numerator becomes:

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

### Step 6: Combine the results
Now we can write \( z \):

\[ z = \frac{-25 + 25i}{25} = -1 + i \]

### Step 7: Find the modulus of \( z \)
The modulus \( |z| \) is calculated as follows:

\[ |z| = \sqrt{(-1)^2 + (1)^2} = \sqrt{1 + 1} = \sqrt{2} \]

### Step 8: Find the argument of \( z \)
The argument \( \theta \) can be found using the formula:

\[ \theta = \tan^{-1}\left(\frac{y}{x}\right) = \tan^{-1}\left(\frac{1}{-1}\right) \]

Since \( z = -1 + i \) lies in the second quadrant, we have:

\[ \theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4} \]

### Step 9: Write the polar form
The polar form of a complex number is given by:

\[ z = r(\cos \theta + i \sin \theta) \]

Thus, substituting the values we found:

\[ z = \sqrt{2} \left( \cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4} \right) \]

### Final Answer
The polar form of the complex number \( z \) is:

\[ z = \sqrt{2} \left( \cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4} \right) \]

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