Answer :
To simplify [tex]\(7i^{22}\)[/tex], we need to understand the properties of the imaginary unit [tex]\(i\)[/tex]. The imaginary unit [tex]\(i\)[/tex] has the property that [tex]\(i^2 = -1\)[/tex].
The powers of [tex]\(i\)[/tex] repeat in a cycle of four, as follows:
- [tex]\(i^1 = i\)[/tex]
- [tex]\(i^2 = -1\)[/tex]
- [tex]\(i^3 = -i\)[/tex]
- [tex]\(i^4 = 1\)[/tex]
After [tex]\(i^4\)[/tex], the cycle starts again, so [tex]\(i^5 = i\)[/tex], [tex]\(i^6 = -1\)[/tex], and so on.
Now, to determine [tex]\(i^{22}\)[/tex], we can find the remainder when 22 is divided by 4. This will help us figure out where 22 lands in the cycle.
1. Divide 22 by 4, which gives a quotient of 5 and a remainder of 2.
Since the remainder is 2, this means that [tex]\(i^{22}\)[/tex] is equivalent to [tex]\(i^2\)[/tex].
We know that [tex]\(i^2 = -1\)[/tex].
Therefore, [tex]\(i^{22} = -1\)[/tex].
Now, let’s find [tex]\(7i^{22}\)[/tex]:
[tex]\(7i^{22} = 7 \times i^{22} = 7 \times (-1) = -7\)[/tex].
So, the simplified form of [tex]\(7i^{22}\)[/tex] is [tex]\(-7\)[/tex].
The powers of [tex]\(i\)[/tex] repeat in a cycle of four, as follows:
- [tex]\(i^1 = i\)[/tex]
- [tex]\(i^2 = -1\)[/tex]
- [tex]\(i^3 = -i\)[/tex]
- [tex]\(i^4 = 1\)[/tex]
After [tex]\(i^4\)[/tex], the cycle starts again, so [tex]\(i^5 = i\)[/tex], [tex]\(i^6 = -1\)[/tex], and so on.
Now, to determine [tex]\(i^{22}\)[/tex], we can find the remainder when 22 is divided by 4. This will help us figure out where 22 lands in the cycle.
1. Divide 22 by 4, which gives a quotient of 5 and a remainder of 2.
Since the remainder is 2, this means that [tex]\(i^{22}\)[/tex] is equivalent to [tex]\(i^2\)[/tex].
We know that [tex]\(i^2 = -1\)[/tex].
Therefore, [tex]\(i^{22} = -1\)[/tex].
Now, let’s find [tex]\(7i^{22}\)[/tex]:
[tex]\(7i^{22} = 7 \times i^{22} = 7 \times (-1) = -7\)[/tex].
So, the simplified form of [tex]\(7i^{22}\)[/tex] is [tex]\(-7\)[/tex].
