https://file.aiquickdraw.com/mathbot/file/1727918284042v9rf1x40.jpg

To solve the division problem \(\frac{-i}{-4 - 5i}\), we will follow these steps:

1. **Multiply both numerator and denominator by the conjugate of the denominator**:  
   The conjugate of \(-4 - 5i\) is \(-4 + 5i\). We do this to eliminate the imaginary part from the denominator.

\[
\frac{-i}{-4 - 5i} \times \frac{-4 + 5i}{-4 + 5i} = \frac{-i(-4 + 5i)}{(-4 - 5i)(-4 + 5i)}
\]

2. **Simplify the denominator**:  
   Using the difference of squares formula for complex numbers:

\[
(-4 - 5i)(-4 + 5i) = (-4)^2 - (5i)^2 = 16 - 25(-1) = 16 + 25 = 41
\]

So the denominator becomes \(41\).

3. **Expand the numerator**:  
   Multiply \(-i\) by \(-4 + 5i\):

\[
-i(-4 + 5i) = 4i - 5(i^2)
\]

Since \(i^2 = -1\), this simplifies to:

\[
4i - 5(-1) = 4i + 5 = 5 + 4i
\]

4. **Final expression**:  
   Now the expression is:

\[
\frac{5 + 4i}{41}
\]

This can be written as:

\[
\frac{5}{41} + \frac{4}{41}i
\]

Thus, the answer in standard form is:

\[
\boxed{\frac{5}{41} + \frac{4}{41}i}
\]

---

Would you like further clarification on any part of this process or more details on how complex number division works? 

Here are some related questions:

1. How do you find the conjugate of a complex number?
2. Why does multiplying by the conjugate help in simplifying complex number division?
3. What are the real and imaginary parts of a complex number?
4. Can you explain the significance of \(i^2 = -1\)?
5. How do you convert complex numbers into polar form?

**Tip:** When dividing complex numbers, always multiply the numerator and denominator by the conjugate of the denominator to simplify the expression.

Divide -i / (-4 - 5i) and write the answer as a complex number in standard form.

Learn how to divide complex numbers by multiplying by the conjugate, and solve the problem (-i) / (-4 - 5i) step-by-step. This example includes simplifying the denominator using the difference of squares and expresses the result in standard form. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation