Answer :
To determine the value of the expression [tex]\(i^0 \times i^1 \times i^2 \times i^3 \times i^4\)[/tex], let's evaluate each term individually where [tex]\(i\)[/tex] is the imaginary unit, defined by [tex]\(i = \sqrt{-1}\)[/tex].
1. Evaluate each power of [tex]\(i\)[/tex]:
- [tex]\(i^0\)[/tex]: Any non-zero number to the power of 0 is 1. So, [tex]\(i^0 = 1\)[/tex].
- [tex]\(i^1\)[/tex]: By definition, [tex]\(i^1 = i\)[/tex].
- [tex]\(i^2\)[/tex]: By definition of the imaginary unit, [tex]\(i^2 = -1\)[/tex].
- [tex]\(i^3\)[/tex]: This can be written as [tex]\(i^2 \times i\)[/tex]. Substituting [tex]\(i^2 = -1\)[/tex], we get [tex]\(i^3 = -1 \times i = -i\)[/tex].
- [tex]\(i^4\)[/tex]: This can be written as [tex]\((i^2)^2\)[/tex]. Substituting [tex]\(i^2 = -1\)[/tex], we get [tex]\(i^4 = (-1)^2 = 1\)[/tex].
2. Multiply the evaluated powers:
Now, we combine the results:
[tex]\[ i^0 \times i^1 \times i^2 \times i^3 \times i^4 = 1 \times i \times -1 \times -i \times 1 \][/tex]
3. Simplify the multiplication step-by-step:
- Start with [tex]\(1 \times i = i\)[/tex]
- Next, [tex]\(i \times -1 = -i\)[/tex]
- Then, [tex]\(-i \times -i = (-1) \times (i \times i) = (-1) \times (i^2) = (-1) \times (-1) = 1\)[/tex]
- Finally, [tex]\(1 \times 1 = 1\)[/tex], but notice there's a mistake. Actually, it should include:
\[
1 \times i \times -1 \times -i \times 1 = i \times -1 \times (-i) = -i \times (-i) = -(-1 \times i \times i) = -(1) = -1
]
So, the value of the expression [tex]\(i^0 \times i^1 \times i^2 \times i^3 \times i^4\)[/tex] is [tex]\(-1\)[/tex].
Therefore, the correct answer is \(\boxed{-1}.
1. Evaluate each power of [tex]\(i\)[/tex]:
- [tex]\(i^0\)[/tex]: Any non-zero number to the power of 0 is 1. So, [tex]\(i^0 = 1\)[/tex].
- [tex]\(i^1\)[/tex]: By definition, [tex]\(i^1 = i\)[/tex].
- [tex]\(i^2\)[/tex]: By definition of the imaginary unit, [tex]\(i^2 = -1\)[/tex].
- [tex]\(i^3\)[/tex]: This can be written as [tex]\(i^2 \times i\)[/tex]. Substituting [tex]\(i^2 = -1\)[/tex], we get [tex]\(i^3 = -1 \times i = -i\)[/tex].
- [tex]\(i^4\)[/tex]: This can be written as [tex]\((i^2)^2\)[/tex]. Substituting [tex]\(i^2 = -1\)[/tex], we get [tex]\(i^4 = (-1)^2 = 1\)[/tex].
2. Multiply the evaluated powers:
Now, we combine the results:
[tex]\[ i^0 \times i^1 \times i^2 \times i^3 \times i^4 = 1 \times i \times -1 \times -i \times 1 \][/tex]
3. Simplify the multiplication step-by-step:
- Start with [tex]\(1 \times i = i\)[/tex]
- Next, [tex]\(i \times -1 = -i\)[/tex]
- Then, [tex]\(-i \times -i = (-1) \times (i \times i) = (-1) \times (i^2) = (-1) \times (-1) = 1\)[/tex]
- Finally, [tex]\(1 \times 1 = 1\)[/tex], but notice there's a mistake. Actually, it should include:
\[
1 \times i \times -1 \times -i \times 1 = i \times -1 \times (-i) = -i \times (-i) = -(-1 \times i \times i) = -(1) = -1
]
So, the value of the expression [tex]\(i^0 \times i^1 \times i^2 \times i^3 \times i^4\)[/tex] is [tex]\(-1\)[/tex].
Therefore, the correct answer is \(\boxed{-1}.
