Powers of the Imaginary Unit $i$ in College Algebra

Powers of the Imaginary Unit i

Definition and Properties of i

The imaginary unit i is defined as the principal square root of -1:

• Definition: $i = \sqrt{-1}$

• All exponent rules for real numbers apply to powers of i.

Many algebraic problems require simplifying expressions involving powers of i, especially for higher exponents.

Fundamental Powers of i

The powers of i repeat in a cycle of four:

• $i^1 = i$

• $i^2 = -1$

• $i^3 = -i$

• $i^4 = 1$

For any integer exponent, the value of $i^n$ can always be simplified to one of these four results: 1, -1, i, -i.

Evaluating Higher Powers of i

To simplify higher powers of i, express the exponent in terms of multiples of 4:

• Any power $i^n$ can be written as $i^{4k + r}$, where $k$ is an integer and $r$ is the remainder when $n$ is divided by 4 ($0 \leq r < 4$).

• The value of $i^n$ depends only on the remainder $r$:

Remainder $r$	Value of $i^n$
0	$i^n = 1$
1	$i^n = i$
2	$i^n = -1$
3	$i^n = -i$

Examples

• Example 1: Simplify $i^{20}$

  • Divide 20 by 4: $20 \div 4 = 5$ remainder 0

  • $i^{20} = i^{4 \times 5 + 0} = 1$

• Example 2: Simplify $i^{22}$

  • Divide 22 by 4: $22 \div 4 = 5$ remainder 2

  • $i^{22} = i^{4 \times 5 + 2} = i^2 = -1$

• Example 3: Simplify $i^{67}$

  • Divide 67 by 4: $67 \div 4 = 16$ remainder 3

  • $i^{67} = i^{4 \times 16 + 3} = i^3 = -i$

Shortcut for Evaluating High Powers of i

To quickly evaluate $i^n$ for large $n$:

1. Divide $n$ by 4 and find the remainder $r$.

2. Use the table above to determine the value of $i^n$.

Practice Problems

• Problem 1: Simplify $i^{1003}$

  • Divide 1003 by 4: $1003 \div 4 = 250$ remainder 3

  • $i^{1003} = i^3 = -i$

• Problem 2: Simplify $i^{85}$

  • Divide 85 by 4: $85 \div 4 = 21$ remainder 1

  • $i^{85} = i^1 = i$

Summary Table: Powers of i

Exponent $n$	Value
$i^1$	$i$
$i^2$	$-1$
$i^3$	$-i$
$i^4$	$1$
$i^5$	$i$
$i^6$	$-1$
$i^7$	$-i$
$i^8$	$1$

Additional info: The concept of powers of i is foundational for working with complex numbers, which are essential in College Algebra and higher mathematics. Mastery of these simplification techniques is crucial for solving equations involving imaginary and complex numbers.