Let's simplify the expression \( i^{4k} \times i^{4k+1} \times i^{4k+2} \times i^{4k+3} \).

Recall that \( i \) is the imaginary unit, where \( i^2 = -1 \) and \( i^4 = 1 \). The powers of \( i \) cycle every four steps:

\[
\begin{aligned}
i^1 & = i, \\
i^2 & = -1, \\
i^3 & = -i, \\
i^4 & = 1.
\end{aligned}
\]

### Step 1: Simplify each power individually
- \( i^{4k} \): Since \( i^4 = 1 \), \( i^{4k} = (i^4)^k = 1^k = 1 \).
- \( i^{4k+1} \): This is equivalent to \( i^{4k} \times i^1 = 1 \times i = i \).
- \( i^{4k+2} \): This is equivalent to \( i^{4k} \times i^2 = 1 \times (-1) = -1 \).
- \( i^{4k+3} \): This is equivalent to \( i^{4k} \times i^3 = 1 \times (-i) = -i \).

### Step 2: Multiply the results
Now, multiply the results of each term:

\[
1 \times i \times (-1) \times (-i) 
\]

Simplifying further:

\[
= i \times (-1) \times (-i) 
= i \times i 
= i^2 = -1
\]

So the result of the expression \( i^{4k} \times i^{4k+1} \times i^{4k+2} \times i^{4k+3} \) is \( \boxed{-1} \).

Would you like more details or have any questions?

Here are 8 related questions you might find interesting:

1. What is the result of \( i^5 \times i^6 \)?
2. How does the cycle of powers of \( i \) repeat after every 4 steps?
3. How can you simplify \( i^{8k+3} \)?
4. What is the geometric interpretation of multiplying complex numbers?
5. How would you express \( i^{4k+5} \times i^{4k+7} \)?
6. Can the powers of \( i \) be represented in polar form?
7. What happens when you multiply \( i \) by a real number?
8. How do you find the roots of a complex number?

**Tip:** Remember that the powers of \( i \) repeat every four terms, which is a helpful shortcut when simplifying expressions involving \( i \).

Learn how to simplify the expression i^(4k) × i^(4k+1) × i^(4k+2) × i^(4k+3) using the properties of the imaginary unit. Follow a detailed step-by-step solution to understand how the powers of i cycle every four terms and result in the answer -1. Suitable for advanced high school students studying complex numbers.

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