To solve the expression [tex]\(\frac{i + i^2 + i^3}{i^3 + i^4 + i^5}\)[/tex], where [tex]\(i = \sqrt{-1}\)[/tex], we can simplify by calculating the powers of [tex]\(i\)[/tex]:
1. Calculate [tex]\(i^2\)[/tex]:
[tex]\[
i^2 = -1
\][/tex]
2. Calculate [tex]\(i^3\)[/tex]:
[tex]\[
i^3 = i \cdot i^2 = i \cdot (-1) = -i
\][/tex]
3. Calculate [tex]\(i^4\)[/tex]:
[tex]\[
i^4 = (i^2)^2 = (-1)^2 = 1
\][/tex]
4. Calculate [tex]\(i^5\)[/tex]:
[tex]\[
i^5 = i \cdot i^4 = i \cdot 1 = i
\][/tex]
Now, plug these into the original expression [tex]\(\frac{i + i^2 + i^3}{i^3 + i^4 + i^5}\)[/tex]:
- Substitute [tex]\(i, i^2,\)[/tex] and [tex]\(i^3\)[/tex] into the numerator:
[tex]\[
i + i^2 + i^3 = i + (-1) + (-i) = i - 1 - i = -1
\][/tex]
- Substitute [tex]\(i^3, i^4,\)[/tex] and [tex]\(i^5\)[/tex] into the denominator:
[tex]\[
i^3 + i^4 + i^5 = -i + 1 + i = 1
\][/tex]
Thus, the original expression simplifies to:
[tex]\[
\frac{-1}{1} = -1
\][/tex]
Therefore, the value of [tex]\(\frac{i + i^2 + i^3}{i^3 + i^4 + i^5}\)[/tex] is [tex]\(-1\)[/tex].
The correct answer is:
[tex]\[ \boxed{-1} \][/tex]
