https://file.aiquickdraw.com/mathbot/file/17297396915360haq2jj2.jpg

To solve \((4 + i)^3\) and express the result in the form \(a + bi\), we can follow these steps:

1. **Expand the expression**:
   \[
   (4 + i)^3 = (4 + i)(4 + i)(4 + i)
   \]
   We first calculate \((4 + i)^2\) and then multiply the result by \(4 + i\).

2. **Calculate \((4 + i)^2\)**:
   \[
   (4 + i)^2 = (4 + i)(4 + i)
   \]
   Expanding:
   \[
   (4 + i)(4 + i) = 4^2 + 2(4)(i) + i^2 = 16 + 8i + i^2
   \]
   Since \(i^2 = -1\), this becomes:
   \[
   16 + 8i - 1 = 15 + 8i
   \]

3. **Multiply the result by \((4 + i)\)**:
   Now multiply \((15 + 8i)\) by \((4 + i)\):
   \[
   (15 + 8i)(4 + i) = 15(4 + i) + 8i(4 + i)
   \]
   Expanding each term:
   \[
   15(4 + i) = 60 + 15i
   \]
   \[
   8i(4 + i) = 32i + 8i^2
   \]
   Since \(i^2 = -1\), the second term becomes:
   \[
   32i + 8(-1) = 32i - 8
   \]

4. **Combine the terms**:
   Now add the real and imaginary parts:
   \[
   60 + 15i + 32i - 8 = (60 - 8) + (15i + 32i) = 52 + 47i
   \]

So, \((4 + i)^3 = 52 + 47i\).

Would you like any further details or clarifications?

Here are 5 related questions you might find useful:
1. How do you simplify powers of other complex numbers, such as \((2 + 3i)^2\)?
2. What is the geometric interpretation of raising complex numbers to powers?
3. How does multiplying complex numbers affect their magnitude and argument?
4. How can you apply De Moivre's Theorem to compute \((4 + i)^3\)?
5. What is the relationship between polar form and powers of complex numbers?

**Tip:** When expanding powers of complex numbers, breaking it down step-by-step using the distributive property helps avoid errors.

This problem demonstrates how to raise the complex number (4 + i) to the third power and simplify the result to the form a + bi. It uses the distributive property and properties of complex numbers, such as i^2 = -1, to reach the solution: 52 + 47i. 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