Math Problem Statement
Simplify each of the following terms involving powers of i into a real or imaginary number, without using any exponents or fractions: i^-5, i^-4, i^-3, i^-2, i^-1, i^0, i^1, i^2, i^3, i^4, i^5.
Solution
To simplify the terms involving powers of , let’s recall that has a cycle of four in its powers. Here’s the cycle:
i^1 &= i, \\ i^2 &= -1, \\ i^3 &= -i, \\ i^4 &= 1. \end{align*}$$ This pattern repeats every four terms. For negative exponents, we use the rule that $$i^{-n} = \frac{1}{i^n}$$ and simplify accordingly. Let's go through each power given in the problem: 1. $$i^{-5}$$: - Using $$i^{-5} = \frac{1}{i^5}$$. - Since $$i^5 = i$$, $$i^{-5} = \frac{1}{i} = -i$$ (by multiplying the numerator and denominator by $$i$$). 2. $$i^{-4}$$: - Using $$i^{-4} = \frac{1}{i^4}$$. - Since $$i^4 = 1$$, $$i^{-4} = \frac{1}{1} = 1$$. 3. $$i^{-3}$$: - Using $$i^{-3} = \frac{1}{i^3}$$. - Since $$i^3 = -i$$, $$i^{-3} = \frac{1}{-i} = i$$. 4. $$i^{-2}$$: - Using $$i^{-2} = \frac{1}{i^2}$$. - Since $$i^2 = -1$$, $$i^{-2} = \frac{1}{-1} = -1$$. 5. $$i^{-1}$$: - Using $$i^{-1} = \frac{1}{i} = -i$$. 6. $$i^0$$: - $$i^0 = 1$$ by definition. 7. $$i^1$$: - $$i^1 = i$$. 8. $$i^2$$: - $$i^2 = -1$$. 9. $$i^3$$: - $$i^3 = -i$$. 10. $$i^4$$: - $$i^4 = 1$$. 11. $$i^5$$: - $$i^5 = i$$. ### Summary Table Here are the simplified results: $$\begin{array}{|c|c|} \hline \text{Power} & \text{Simplified Result} \\ \hline i^{-5} & -i \\ i^{-4} & 1 \\ i^{-3} & i \\ i^{-2} & -1 \\ i^{-1} & -i \\ i^0 & 1 \\ i^1 & i \\ i^2 & -1 \\ i^3 & -i \\ i^4 & 1 \\ i^5 & i \\ \hline \end{array}$$ Would you like more details on any of these steps or explanations for handling powers of $$i$$? --- ### Related Questions 1. What are the general rules for simplifying negative powers of imaginary numbers? 2. How do we derive the recurring cycle in powers of $$i$$? 3. Why does the pattern in powers of $$i$$ repeat every four terms? 4. How can this knowledge of $$i$$'s powers apply to complex number calculations? 5. What is the impact of imaginary numbers in practical applications like engineering? ### Tip When working with powers of $$i$$, remember that simplifying to the lowest positive equivalent exponent within the four-term cycle will make calculations much easier.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Complex numbers
Imaginary unit
Exponents
Patterns in powers of i
Formulas
i^1 = i
i^2 = -1
i^3 = -i
i^4 = 1
For negative exponents: i^-n = 1 / i^n
Theorems
Cycle of powers of i repeats every four terms
Suitable Grade Level
High School (Grades 9-12)