https://file.aiquickdraw.com/mathbot/file/17262582870855z2yg4zy.jpg

The expression shown in the image is:

\[
\frac{4}{(5 + i)(6 - i)}
\]

To simplify this, we'll follow these steps:

### Step 1: Multiply the denominator by its conjugate
We will multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of \((6 - i)\) is \((6 + i)\).

\[
\frac{4}{(5 + i)(6 - i)} \times \frac{6 + i}{6 + i}
\]

### Step 2: Expand both the numerator and denominator

- **Numerator:**

\[
4 \times (6 + i) = 24 + 4i
\]

- **Denominator:**

Using the difference of squares formula: \((a - b)(a + b) = a^2 - b^2\)

\[
(5 + i)(6 - i) = (5 \times 6) + (5 \times -i) + (i \times 6) + (i \times -i)
= 30 - 5i + 6i - i^2
\]
Since \(i^2 = -1\):

\[
30 - 5i + 6i + 1 = 31 + i
\]

### Step 3: Final Expression

Now, the simplified expression is:

\[
\frac{24 + 4i}{31 + i}
\]

This is the simplified version. Would you like to proceed further with additional simplification, such as rationalizing the denominator, or are you satisfied with this form?

Here are five related questions you could explore:

1. What is the purpose of multiplying by the conjugate when dealing with complex numbers?
2. How do you simplify an expression when the denominator contains complex numbers?
3. What is the geometric interpretation of complex numbers?
4. Can this expression be further simplified or represented differently?
5. How does the difference of squares formula help in simplifying complex numbers?

**Tip:** When working with complex numbers, always check if simplifying through conjugates can help clear out imaginary parts from the denominator.

Simplify the expression 4 / ((5 + i)(6 - i))

Learn how to simplify the complex number expression 4 / ((5 + i)(6 - i)) by multiplying the denominator by its conjugate. This solution covers key algebraic concepts like difference of squares and complex numbers. 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