Math Problem Statement
Simplify i^9.
Solution
To simplify , where is the imaginary unit (), let's follow these steps:
Step 1: Identify the pattern of powers of :
The powers of repeat every 4 cycles as follows:
i^1 &= i, \\ i^2 &= -1, \\ i^3 &= -i, \\ i^4 &= 1. \end{aligned}$$ This cycle ($$i, -1, -i, 1$$) repeats for higher powers of $$i$$. --- ### Step 2: **Find the remainder when 9 is divided by 4:** To determine the position of $$i^9$$ in the cycle, compute $$9 \div 4$$: $$9 \div 4 = 2 \text{ remainder } 1.$$ Thus, $$9$$ can be written as: $$9 = 4 \cdot 2 + 1.$$ This means $$i^9$$ is equivalent to $$i^1$$. --- ### Step 3: **Simplify:** From the cycle of powers, $$i^1 = i$$. Therefore: $$i^9 = i.$$ --- ### Final Answer: $$i^9 = i.$$ Would you like further details or a deeper explanation of the powers of $$i$$? --- ### Related Questions: 1. Simplify $$i^{12}$$ using the same method. 2. What is the value of $$i^{15}$$? 3. Explain the pattern of powers of $$i$$ and why it repeats every 4 steps. 4. Solve $$(i^2 + i^3) \times i^4$$. 5. Determine the value of $$i^{2023}$$. ### Tip: To simplify any power of $$i$$, always divide the exponent by 4 and use the remainder to find the position in the cycle $$(i, -1, -i, 1)$$.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Complex Numbers
Powers of Imaginary Unit
Formulas
i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1
Theorems
Cyclic Nature of Powers of i
Suitable Grade Level
Grades 9-12