i³ equals -i because it stems from the definition of i itself, where i represents the imaginary unit, and i² = -1.
Explanation
The imaginary unit i is defined as the square root of -1. Therefore, when we deal with powers of i, we can simplify them using the fundamental property that i² = -1.
Here's a breakdown of how i³ simplifies to -i:
-
Express i³ as a product: We can rewrite i³ as i² i*.
-
Substitute i² with -1: Since i² = -1, we substitute -1 for i² in the expression. So, i³ = (-1) i*.
-
Simplify: Multiplying -1 by i simply gives us -i. Thus, i³ = -i.
Table Demonstrating Powers of i
| Power of i | Simplification | Result |
|---|---|---|
| i¹ | i | i |
| i² | -1 | -1 |
| i³ | i² * i | -i |
| i⁴ | i² i² = (-1) (-1) | 1 |
Key Concept
The simplification relies heavily on the definition of i². It is important to note that the rule √a√b=√ab is only guaranteed for positive real a and b. Applying this rule incorrectly when dealing with complex numbers can lead to errors. Instead, always revert to the fundamental definition of i² = -1 when simplifying expressions involving the imaginary unit.
