Math Problem Statement

Simplify the following expression:

2−i3+2i2−i3+2i

Solution

Let's simplify the expression:

2i3+2i2i3+2i2 - i3 + 2i2 - i3 + 2i

Step-by-step breakdown:

1. Clarify terms:

It seems there may be some formatting confusion. I interpret it as:

2i3+2i2i3+2i2 - i \cdot 3 + 2i \cdot 2 - i \cdot 3 + 2i

Which means we have:

  • 22 is a real constant.
  • i3- i \cdot 3 is a complex term, where ii is the imaginary unit.
  • 2i22i \cdot 2 is also a complex term.
  • Another i3-i \cdot 3 and +2i+ 2i.

2. Simplify the individual terms:

  • i3=3i-i \cdot 3 = -3i
  • 2i2=4i2i \cdot 2 = 4i
  • The second i3=3i-i \cdot 3 = -3i
  • The last +2i+ 2i remains as it is.

3. Combine like terms:

Now, putting everything together:

23i+4i3i+2i2 - 3i + 4i - 3i + 2i

4. Combine the real terms:

  • The real part is simply 22.

5. Combine the imaginary terms:

  • 3i+4i3i+2i=0i-3i + 4i - 3i + 2i = 0i.

Thus, the entire expression simplifies to:

22

Final Answer:

The simplified expression is 2\boxed{2}.


Would you like further clarification or a breakdown of any specific steps?

Here are 5 related questions for you:

  1. What is the imaginary unit ii, and how does it behave in mathematical expressions?
  2. How do you simplify expressions involving complex numbers?
  3. Can you explain the concept of real and imaginary parts of a complex number?
  4. What is the geometric interpretation of complex numbers in the complex plane?
  5. How do you add and subtract complex numbers?

Tip: Always group and simplify real and imaginary parts separately when dealing with complex numbers.

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Math Problem Analysis

Mathematical Concepts

Complex Numbers
Simplification of Expressions
Real and Imaginary Parts

Formulas

i = √-1
Real and Imaginary Parts of Complex Numbers

Theorems

Simplification of Complex Expressions

Suitable Grade Level

Grades 9-11