Find the values of the real numbers 𝑥 and 𝑦 if: (3𝑦 − 2)𝑖¹⁶ + (5 − 2𝑥)𝑖 = 0

We are given the equation:

\[
(3y - 2)i^{16} + (5 - 2x)i = 0
\]

Here, \( i \) is the imaginary unit, which satisfies \( i^2 = -1 \). We need to find the real values of \( x \) and \( y \).

### Step 1: Simplify Powers of \( i \)
First, let's simplify the powers of \( i \):

- \( i^{16} \): Powers of \( i \) cycle every four terms: \( i^1 = i \), \( i^2 = -1 \), \( i^3 = -i \), and \( i^4 = 1 \). Since \( 16 \) is a multiple of 4, \( i^{16} = 1 \).

Thus, the equation becomes:
\[
(3y - 2) \cdot 1 + (5 - 2x)i = 0
\]
Simplifying further:
\[
3y - 2 + (5 - 2x)i = 0
\]

### Step 2: Set Up Real and Imaginary Parts
For this equation to be true, both the real part and the imaginary part must equal zero.

- **Real part**: \( 3y - 2 = 0 \)
- **Imaginary part**: \( 5 - 2x = 0 \)

### Step 3: Solve for \( x \) and \( y \)
1. **Solving for \( y \)**:
   \[
   3y - 2 = 0 \quad \Rightarrow \quad 3y = 2 \quad \Rightarrow \quad y = \frac{2}{3}
   \]

2. **Solving for \( x \)**:
   \[
   5 - 2x = 0 \quad \Rightarrow \quad 2x = 5 \quad \Rightarrow \quad x = \frac{5}{2}
   \]

### Final Answer
The real values of \( x \) and \( y \) are:
\[
x = \frac{5}{2}, \quad y = \frac{2}{3}
\]

Would you like further details, or do you have any questions?

Here are 5 related questions:
1. How does the cyclic nature of \( i \) affect solving complex equations?
2. What is the general form of solving equations with real and imaginary parts?
3. How would this equation change if we had \( i^8 \) instead of \( i^{16} \)?
4. What are the properties of complex numbers when equated to zero?
5. How can we represent complex numbers geometrically on the complex plane?

**Tip**: When dealing with powers of \( i \), always reduce the exponent mod 4 to simplify calculations.

Explore how to solve the complex equation (3y - 2)i^16 + (5 - 2x)i = 0 for real numbers x and y. Learn about the properties of complex numbers, powers of the imaginary unit, and step-by-step solution methods. Suitable for advanced high school students and beyond.

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