https://r2.erweima.ai/mathbot/image/172599013492858jo1te2.jpg

The equation provided in the image is:

\[
\frac{1}{z} = \frac{-4 + 8i}{8i}
\]

We are tasked with determining the value of \( z \).

### Step-by-step solution:

1. Simplify the right-hand side:
   \[
   \frac{-4 + 8i}{8i}
   \]

   We can break this into two parts:
   \[
   \frac{-4}{8i} + \frac{8i}{8i}
   \]

   This simplifies to:
   \[
   \frac{-4}{8i} + 1 = \frac{-1}{2i} + 1
   \]

2. Simplify \( \frac{-1}{2i} \) by multiplying numerator and denominator by \( i \) to eliminate the imaginary unit in the denominator:
   \[
   \frac{-1}{2i} \cdot \frac{i}{i} = \frac{-i}{2(-1)} = \frac{i}{2}
   \]

   So, we now have:
   \[
   \frac{1}{z} = \frac{i}{2} + 1 = 1 + \frac{i}{2}
   \]

3. Now, take the reciprocal of both sides to find \( z \):
   \[
   z = \frac{1}{1 + \frac{i}{2}}
   \]

4. To simplify this, multiply both the numerator and the denominator by the conjugate of the denominator \( 1 - \frac{i}{2} \):
   \[
   z = \frac{1 \cdot (1 - \frac{i}{2})}{(1 + \frac{i}{2})(1 - \frac{i}{2})}
   \]

   The denominator simplifies as follows:
   \[
   (1 + \frac{i}{2})(1 - \frac{i}{2}) = 1^2 - \left(\frac{i}{2}\right)^2 = 1 - \frac{-1}{4} = 1 + \frac{1}{4} = \frac{5}{4}
   \]

   So, we have:
   \[
   z = \frac{1 - \frac{i}{2}}{\frac{5}{4}} = \frac{4(1 - \frac{i}{2})}{5} = \frac{4 - 2i}{5}
   \]

Thus, the value of \( z \) is:

\[
z = \frac{4}{5} - \frac{2i}{5}
\]

### Final Answer:
\[
z = \frac{4}{5} - \frac{2i}{5}
\]

---

Would you like further clarification or additional details?

Here are 5 related questions for you:
1. How do you find the conjugate of a complex number?
2. What is the general process for simplifying complex fractions?
3. Why do we multiply by the conjugate when dividing complex numbers?
4. How can we represent the solution \( z = \frac{4}{5} - \frac{2i}{5} \) in polar form?
5. How would you solve this problem using an alternative method?

**Tip**: When simplifying complex fractions, always try to eliminate the imaginary unit from the denominator by multiplying by the conjugate!

Si 1/z = (-4 + 8i) / 8i, determina el valor de z.

Step-by-step solution to find the value of z in the complex equation 1/z = (-4 + 8i) / 8i. This problem involves complex number simplifications, the use of conjugates, and reciprocal operations. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation