https://r2.erweima.ai/mathbot/image/17242086306168c2iptr6.jpg

resuelva el siguiente problema de cálculo nivel 2, operaciones de números complejos según corresponda. Debe simplificar al máximo la operación 

Let's solve the given complex number expression step by step. The expression is:

\[
\frac{2i^{113}(2 - i^{2024}) - (3 + 3i)}{(2 + 5i) + (2 + 6i)}
\]

### Step 1: Simplify the Exponentiated Imaginary Numbers
First, simplify \(i^{113}\) and \(i^{2024}\).

Remember that the powers of \(i\) cycle every four terms:
\[
i^1 = i, \quad i^2 = -1, \quad i^3 = -i, \quad i^4 = 1
\]

#### For \(i^{113}\):
Find the remainder when 113 is divided by 4:
\[
113 \div 4 = 28 \text{ R } 1
\]
So, \(i^{113} = i^1 = i\).

#### For \(i^{2024}\):
Find the remainder when 2024 is divided by 4:
\[
2024 \div 4 = 506 \text{ R } 0
\]
So, \(i^{2024} = i^0 = 1\).

### Step 2: Substitute the Simplified Values
Now, substitute \(i^{113} = i\) and \(i^{2024} = 1\) into the expression:
\[
\frac{2i(2 - 1) - (3 + 3i)}{(2 + 5i) + (2 + 6i)}
\]
This simplifies to:
\[
\frac{2i(1) - (3 + 3i)}{(4 + 11i)}
\]

### Step 3: Simplify the Numerator
Expand and simplify the numerator:
\[
2i(1) - (3 + 3i) = 2i - 3 - 3i = -3 - i
\]

### Step 4: Combine the Results
The expression now becomes:
\[
\frac{-3 - i}{4 + 11i}
\]

### Step 5: Rationalize the Denominator
To simplify the expression, multiply the numerator and the denominator by the conjugate of the denominator, which is \(4 - 11i\):
\[
\frac{(-3 - i)(4 - 11i)}{(4 + 11i)(4 - 11i)}
\]

First, compute the denominator:
\[
(4 + 11i)(4 - 11i) = 4^2 - (11i)^2 = 16 - (-121) = 137
\]

Now, compute the numerator:
\[
(-3 - i)(4 - 11i) = -3(4 - 11i) - i(4 - 11i) = (-12 + 33i) - (4i - 11) = -12 + 33i - 4i + 11 = -1 + 29i
\]

Thus, the expression becomes:
\[
\frac{-1 + 29i}{137}
\]

### Step 6: Final Answer
Finally, simplify by separating the real and imaginary parts:
\[
\text{Simplified result} = \frac{-1}{137} + \frac{29}{137}i
\]

This is the simplest form of the expression.

Would you like any further details or clarifications?

### Relative Questions
1. How does the cycle of powers of \(i\) help in simplifying expressions involving complex numbers?
2. What is the significance of multiplying by the conjugate in complex number operations?
3. Can you explain the impact of rationalizing the denominator on the simplification process?
4. What are other methods to simplify expressions involving imaginary numbers?
5. How can complex numbers be represented graphically on the complex plane?

### Tip
Remember that the powers of \(i\) cycle every four steps, which is a handy tool for quickly simplifying complex expressions involving high powers of \(i\).

Learn how to simplify a complex number expression involving high powers of i. This problem covers operations with complex numbers, including simplifying using the cycle of powers of i and rationalizing the denominator. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation