To solve for [tex]\(i^3\)[/tex] using the properties of imaginary numbers and exponents, follow these steps:
1. Understanding [tex]\(i\)[/tex] and its powers:
- Recall that [tex]\(i\)[/tex] is the imaginary unit, defined by [tex]\(i^2 = -1\)[/tex].
2. Calculate [tex]\(i^2\)[/tex]:
- By definition, [tex]\(i^2 = -1\)[/tex].
3. Exponentiation Law:
- Use the property of exponents: [tex]\(i^3 = (i^2) \cdot i\)[/tex].
4. Calculate [tex]\(i^3\)[/tex]:
- Substitute [tex]\(i^2 = -1\)[/tex] into the expression for [tex]\(i^3\)[/tex]:
[tex]\[
i^3 = (i^2) \cdot i = (-1) \cdot i = -i
\][/tex]
5. Imaginary Number Form:
- Writing in the form of complex numbers, where the real part is 0:
[tex]\[
-i = 0 - 1i
\][/tex]
So, the calculations give us:
- [tex]\(i^2\)[/tex] is [tex]\(-1\)[/tex]
- [tex]\(i^3\)[/tex] is [tex]\(-i\)[/tex] or represented as [tex]\(0 - 1i\)[/tex].
Thus, we have:
[tex]\[
i^2 = -1 + 0j
\][/tex]
[tex]\[
i^3 = -0 - 1j
\][/tex]
These results match our understanding of the powers of the imaginary unit [tex]\(i\)[/tex].
