If z=3+4i, then find |z| and `z^(-1)`.

Answer

To solve the problem step by step, we will find the modulus of the complex number \( z = 3 + 4i \) and then find its inverse \( z^{-1} \).

### Step 1: Find the modulus of \( z \)

The modulus of a complex number \( z = a + bi \) is given by the formula:

\[ |z| = \sqrt{a^2 + b^2} \]

For \( z = 3 + 4i \):
- Here, \( a = 3 \) and \( b = 4 \).

Now, we can calculate:

\[ |z| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 2: Find the inverse of \( z \)

The inverse of a complex number \( z \) is given by:

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

To simplify this expression, we will multiply the numerator and the denominator by the conjugate of the denominator:

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

Now, we need to calculate the denominator:

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

Thus, we have:

\[ z^{-1} = \frac{3 - 4i}{25} = \frac{3}{25} - \frac{4}{25}i \]

### Final Results

1. The modulus of \( z \) is \( |z| = 5 \).
2. The inverse of \( z \) is \( z^{-1} = \frac{3}{25} - \frac{4}{25}i \).

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