To solve the equation \( (\sqrt{2} - i) - i(1 - \sqrt{2}i) = -2i \), let's proceed step by step.

### Step 1: Expand the expression
Start by expanding the term \( -i(1 - \sqrt{2}i) \):

\[
-i(1 - \sqrt{2}i) = -i \cdot 1 + (-i) \cdot (-\sqrt{2}i)
\]

\[
= -i + \sqrt{2}i^2
\]

Since \( i^2 = -1 \), we have:

\[
\sqrt{2}i^2 = \sqrt{2}(-1) = -\sqrt{2}
\]

Thus:

\[
-i(1 - \sqrt{2}i) = -i - \sqrt{2}
\]

### Step 2: Substitute into the original equation
Now substitute this back into the original expression:

\[
(\sqrt{2} - i) - (-i - \sqrt{2}) = -2i
\]

This simplifies to:

\[
\sqrt{2} - i + i + \sqrt{2} = -2i
\]

### Step 3: Combine like terms
Combine the real and imaginary parts:

\[
(\sqrt{2} + \sqrt{2}) + (-i + i) = -2i
\]

\[
2\sqrt{2} + 0 = -2i
\]

### Step 4: Conclusion
This simplifies to:

\[
2\sqrt{2} = -2i
\]

This equation is not true, as the left-hand side is a real number while the right-hand side is purely imaginary. Therefore, the given equation does **not** hold true.

Would you like further details on any part of this process or have any questions?

---

Here are some related questions to consider:

1. What are the properties of imaginary numbers and complex conjugates?
2. How do you add and subtract complex numbers?
3. What is the significance of the imaginary unit \( i \) and its powers?
4. How do you multiply complex numbers?
5. What are some common mistakes when manipulating complex numbers?

**Tip:** When dealing with complex numbers, always be careful to distinguish between real and imaginary parts, as they are not interchangeable.

Explore how to solve the equation (sqrt(2) - i) - i(1 - sqrt(2)i) = -2i step-by-step. This problem delves into complex number operations and the distinction between real and imaginary components. Suitable for advanced high school students and beyond.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation