Powers of i

TipLearning Objectives

By the end of this lesson, you’ll be able to:

  • Use the definition \(i^2 = -1\) to simplify powers of \(i\).
  • Recognize and apply the repeating cycle of \(i\).
  • Reduce any power of \(i\) using modular arithmetic.
  • Rewrite expressions involving \(i^n\) in simplest form.

Key Ideas

Imaginary unit: \[ i = \sqrt{-1} \]

The key identity: \[ i^2 = -1 \]

Using this, we find: \[ i^3 = i^2 \cdot i = -1 \cdot i = -i \]

\[ i^4 = (i^2)^2 = (-1)^2 = 1 \]

Then the cycle repeats every 4 powers:

Power Value
\(i^1\) \(i\)
\(i^2\) \(-1\)
\(i^3\) \(-i\)
\(i^4\) \(1\)

And then: - \(i^5 = i\)
- \(i^6 = -1\)
- \(i^7 = -i\)
- \(i^8 = 1\)
…and so on.

Shortcut

To simplify \(i^n\):

  1. Compute \(n \bmod 4\).
  2. Match the remainder to the cycle:
\(n \bmod 4\) Result
1 \(i\)
2 \(-1\)
3 \(-i\)
0 \(1\)

Common Problem Types

Simplifying Large Powers

Use the cycle or modulo.

Example:
\(i^{23}\)
Compute \(23 \bmod 4 = 3\) → result = \(-i\).

Negative Exponents

Rewrite using reciprocals.

Example:
\[ i^{-3} = \frac{1}{i^3} = \frac{1}{-i} = i \]

Combining Powers

Simplify each part first.

Example:
\[ i^7 + i^{12} \]
\(i^7 = -i\) (since \(7 \bmod 4 = 3\))
\(i^{12} = 1\) (since \(12 \bmod 4 = 0\))
Result: \(1 - i\)

Using \(i^2 = -1\) Inside Algebra

Convert \(i^2\) to \(-1\) to simplify expressions.

Example:
\[ 3i^2 - 5i = 3(-1) - 5i = -3 - 5i \]

Strategies

  • Always reduce exponents modulo 4.
  • Replace \(i^2\) with \(-1\) immediately.
  • Look for patterns in long expressions.
  • Treat \(i\) algebraically — combine like terms.

Worked Examples

Example 1 — Simplify \(i^{37}\)

Compute \(37 \bmod 4 = 1\).
So: \[ i^{37} = i. \]

Example 2 — Simplify \(i^{14}\)

Compute \(14 \bmod 4 = 2\).
So: \[ i^{14} = -1. \]

Example 3 — Simplify an expression

\[ 2i^3 - 4i^5 \]

Compute each:
\(i^3 = -i\)
\(i^5 = i\)

So: \[ 2(-i) - 4(i) = -2i - 4i = -6i. \]

WarningCommon Mistakes
  • Forgetting the cycle every 4 powers.
  • Thinking powers grow in magnitude — they don’t, they loop.
  • Incorrectly simplifying negative exponents.
  • Mixing up \(i^3 = -i\) (not \(i\)).

Practice Problems

  1. Simplify \(i^{52}\).
  2. Simplify \(i^{15}\).
  3. Evaluate \(i^{-1}\).
  4. Simplify \(3i^2 + 5i^3\).
  5. Simplify \(i^{27} + i^{100}\).

1. \(52 \bmod 4 = 0\) → \(i^{52} = 1\)

2. \(15 \bmod 4 = 3\) → \(i^{15} = -i\)

3.
\[ i^{-1} = \frac{1}{i} = -i \]
(rationalizing denominator)

4.
\(3i^2 = 3(-1) = -3\)
\(5i^3 = 5(-i) = -5i\)
So: \(-3 - 5i\)

5.
\(i^{27}\): \(27 \bmod 4 = 3\) → \(-i\)
\(i^{100}\): \(100 \bmod 4 = 0\) → \(1\)
Sum: \(1 - i\)

Summary

  • Powers of \(i\) repeat every 4.
  • Reduce exponents using \(n \bmod 4\).
  • \(i^2 = -1\) is the key identity.
  • Simplify expressions by converting powers and combining like terms.
  • Memorize the cycle: \(i, -1, -i, 1\).
  • Use mod 4 for any power.
  • Negative exponents → reciprocals.
  • Replace \(i^2\) with \(-1\) ASAP.