Questions: Simplify the powers of i. Part 1 of 4 i^-22=

Transcript text: Simplify the powers of $i$. Part 1 of 4 \[ i^{-22}= \]

Solution

Solution Steps

To simplify the powers of $i$, we need to use the cyclical nature of the powers of the imaginary unit $i$. The powers of $i$ repeat every four terms: $i, -1, -i, 1$. We can use this property to simplify $i^{-22}$.

Find the equivalent positive exponent by adding multiples of 4 until the exponent is positive.
Reduce the exponent modulo 4 to find the equivalent exponent within the first cycle (0 to 3).
Use the known values of $i^0, i^1, i^2,$ and $i^3$ to determine the result.

Step 1: Convert to Positive Exponent

To simplify $i^{-22}$, we first convert the negative exponent to a positive one. We do this by finding the equivalent positive exponent using modulo 4: \[ -22 \mod 4 = 2 \]

Step 2: Identify the Power of $i$

Next, we identify the value of $i^2$ using the cyclical nature of the powers of $i$: \[ i^0 = 1, \quad i^1 = i, \quad i^2 = -1, \quad i^3 = -i \] From this, we see that: \[ i^2 = -1 \]

Final Answer

Thus, the simplified form of $i^{-22}$ is: \[ \boxed{-1} \]

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