Question
Perform the indicated operation, and write the expression. i^20+i^10-i^4-i^2
Ask by Klein Fowler.
Mar 31,2025 00:09
UpStudy AI Solution
Tutor-Verified Answer
Answer
The expression simplifies to 0.
Solution
Calculate or simplify the expression \( i^{20}+i^{10}-i^{4}-i^{2} \).
Simplify the expression by following steps:
- step0: Solution:
\(i^{20}+i^{10}-i^{4}-i^{2}\)
- step1: Evaluate the power:
\(1+i^{10}-i^{4}-i^{2}\)
- step2: Evaluate the power:
\(1-1-i^{4}-i^{2}\)
- step3: Evaluate the power:
\(1-1-1-i^{2}\)
- step4: Evaluate the power:
\(1-1-1-\left(-1\right)\)
- step5: Simplify:
\(-1-\left(-1\right)\)
- step6: Subtract the terms:
\(0\)
The expression \(i^{20}+i^{10}-i^{4}-i^{2}\) simplifies to 0.
Answered by UpStudy AI and reviewed by a Professional Tutor
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Simplify this solution Bonus Knowledge
Let's simplify the expression step by step.
First, recall that \( i \) is the imaginary unit and \( i^2 = -1 \). The powers of \( i \) follow a cyclical pattern:
- \( i^1 = i \)
- \( i^2 = -1 \)
- \( i^3 = -i \)
- \( i^4 = 1 \)
- \( i^5 = i \) (and it repeats)
Now we can simplify each term in the expression:
1. \( i^{20} = (i^4)^5 = 1^5 = 1 \)
2. \( i^{10} = (i^4)^2 \cdot i^2 = 1^2 \cdot (-1) = -1 \)
3. \( -i^{4} = -1 \)
4. \( -i^{2} = -(-1) = 1 \)
Now, substitute these values back into the expression:
\[
i^{20} + i^{10} - i^{4} - i^{2} = 1 + (-1) - 1 + 1
\]
Now, simplify:
\[
1 - 1 - 1 + 1 = 0
\]
So the final result is:
\[
\boxed{0}
\]
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