1. Equations & Inequalities
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Simplify the power of .
A
B
-1
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D
1
Verified step by step guidance1
Identify the pattern in the powers of the imaginary unit \(i\). Recall that \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\). This pattern repeats every four powers.
To simplify \(i^{85}\), determine the remainder when 85 is divided by 4, since the powers of \(i\) repeat every 4.
Perform the division: 85 divided by 4 gives a quotient and a remainder. Focus on the remainder, as it will determine the equivalent power of \(i\).
The remainder from the division will tell you which power of \(i\) corresponds to \(i^{85}\). For example, if the remainder is 1, then \(i^{85} = i^1 = i\).
Use the remainder to match with the known powers of \(i\): \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), or \(i^4 = 1\). This will give you the simplified form of \(i^{85}\).
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