Practice Objectives
- Demonstrate an understanding of the repeating cycle of powers of i
- Demonstrate the ability to simplify any integer power of i
Practice Simplifying Powers of the Imaginary Unit
Instructions:
Answer 7/10 questions correctly to pass.
Find each power of i and then choose the simplified form as 1, i, -1, or -i.
Note: Since i represents the square root of -1, you must algebraically eliminate i from any denominator.
Problem:
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Simplifying Powers of i:
- If the exponent on i is negative, rewrite the power of i using the rule for negative exponents:
- $$i^{-a} = \frac{1}{i^{a}}, \, a > 0$$
- After rewriting, simplify the denominator
- If the exponent on i is 0-4:
- $$i^0 = 1$$
- $$i^1 = i$$
- $$i^2 = -1$$
- $$i^3 = -i$$
- $$i^4 = 1$$
- If the exponent on i is larger than 4, divide the exponent by 4 and record the remainder
- Use the remainder to find the simplified value:
- The remainder will be either 0, 1, 2, or 3
- Match up the remainder with the corresponding exponent from the cycle (listed above)
- Algebraically eliminate i from any denominator:
- Multiply both numerator and denominator by i to eliminate i from the denominator
Step-by-Step:
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