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All courses Math II · N-CN.1 55 of 73
Understand i as a number with i^2=-1 and represent complex numbers as a+bi.

Simplify powers of $i$ using the repeating cycle $i, -1, -i, 1$

Problem
Evaluate i⁵ by writing 5=4+1 and using i⁴=1. State the resulting value from the four-power cycle.
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Hint

Recall the repeating cycle of powers of \(i\): \(i, -1, -i, 1\).

The powers of \(i\) repeat every 4, so reduce the exponent modulo 4.

Solution walkthrough

01

Recall the four-power cycle

\[i^1=i;~i^2=-1;~i^3=-i;~i^4=1\]

Powers of i repeat every four exponents because multiplying i⁴=1 by another cycle leaves the value unchanged.

02

Decompose the exponent

\[5=4+1\]

Separating a full cycle from the remainder exposes the same value as exponent 1.

03

Apply exponent rules

\[i^5=i^(4+1)=i^4*i=1*i=i\]

For the same base, adding exponents corresponds to multiplication.

04

Check by residue

\[5~\mod~4=1~->~\text{same}~\text{cycle}~\text{position}~\text{as}~i^1\]

The remainder 1 independently confirms the result i.

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Another way

  1. Expand i⁵=i*i*i*i*i=(i²)(i²)i=(-1)(-1)i=i.

!

Common mistake

Do not multiply the exponent by i. Exponents select a position in the repeating four-value cycle.

Answer power-cycle diagram: powers of i repeat 1, i, negative 1, negative i every four exponents. Since 5 divided by 4 has remainder 1, the residue-1 position i is highlighted, so i to the fifth equals i.
The modulo-four cycle explains the exponent reduction and highlights the landing value after submission.