Math Problem Statement
I^59
Solution
To simplify , we can use the properties of powers of the imaginary unit .
Key Property of
The imaginary unit satisfies:
From this, we can find a repeating pattern for powers of :
i^1 &= i, \\ i^2 &= -1, \\ i^3 &= -i, \\ i^4 &= 1. \end{align*}$$ After $$i^4 = 1$$, the powers repeat every 4 steps. This means the powers of $$i$$ form a cycle of 4: $$i, -1, -i, 1, i, -1, -i, 1, \ldots$$. ### Simplifying $$i^{59}$$ To simplify $$i^{59}$$, we find the remainder when 59 is divided by 4, since the powers repeat every 4. $$59 \div 4 = 14 \text{ remainder } 3$$ So, $$59 \equiv 3 \pmod{4}$$. This means: $$i^{59} = i^3$$ From the pattern we know: $$i^3 = -i$$ ### Final Answer $$i^{59} = -i$$ Would you like to explore more about powers of $$i$$ or have any questions? Here are 5 related questions to deepen your understanding: 1. How would you simplify $$i^{101}$$? 2. Why does the power of $$i$$ repeat every 4 terms? 3. How do complex exponents differ from real exponents? 4. Can you visualize the powers of $$i$$ on the complex plane? 5. What happens if you raise $$i$$ to a negative power? **Tip:** Remember, finding the remainder when dividing by 4 is the key to simplifying powers of $$i$$.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Complex Numbers
Imaginary Unit
Exponentiation
Formulas
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Theorems
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Suitable Grade Level
Grades 10-12