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Algebra 2 > K. 8 Powers of I EUT Which number equals \( i^{9} \) ? W \( -i \square 1 \)

Ask by Hilton Logan. in the United States
Jan 10,2025

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Answer

\( i^9 = i \)

Solution

To determine the value of \( i^9 \), let's analyze the powers of the imaginary unit \( i \). The powers of \( i \) cycle every four exponents: \[ \begin{align*} i^1 &= i \\ i^2 &= -1 \\ i^3 &= -i \\ i^4 &= 1 \\ \end{align*} \] This cycle repeats every four powers. To find \( i^9 \), divide the exponent by 4 and look at the remainder: \[ 9 \div 4 = 2 \text{ with a remainder of } 1 \] This means: \[ i^9 = i^{4 \times 2 + 1} = (i^4)^2 \times i^1 = 1^2 \times i = i \] **Answer:** \( i^9 = i \)

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Mind Expander

To determine \( i^9 \), we first need to recall that the powers of \( i \) (the imaginary unit) cycle every four: \( i^1 = i \), \( i^2 = -1 \), \( i^3 = -i \), and \( i^4 = 1 \). Since \( 9 \mod 4 = 1 \), we find that \( i^9 = i^1 = i \). Now, evaluating \( i^{n} \) allows us to easily see its periodic nature, which can simplify complex calculations. The key is recognizing that any power of \( i \) can be reduced by using modular arithmetic with 4, saving time while ensuring accuracy! It’s a fun little trick that keeps everything organized and predictable!

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