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	Solving Systems of Equations - Two Lines
	Solving Nonlinear Equations by Factoring
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	Rationalizing the Denominator
	Simplifying Complex Fractions
	Factoring Trinomials
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	Axis of Symmetry and Vertices
	Equations Quadratic in Form
	The Appearance of a Polynomial Equation
	Subtracting Reverses
	Non-Linear Equations
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	Factoring Trinomials by Grouping
	Factoring Trinomials of the Type ax 2 + bx + c
	The Distance Formula
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	Solving a System of Three Linear Equations by Elimination
	Multiplication by 25
	Powers of i
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	Slope-intercept Form for the Equation of a Line
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	Product Rule for Radicals
	Solving Compound Linear Inequalities
	Axis of Symmetry and Vertices
	Multiplying Rational Expressions
	Reducing Rational Expressions
	Properties of Negative Exponents
	Fractions
	Numbers, Factors, and Reducing Fractions to Lowest Terms
	Solving Quadratic Equations
	Factoring Completely General Quadratic Trinomials
	Solving a Formula for a Given Variable
	Factoring Polynomials
	Decimal Numbers and Fractions
	Multiplication Properties of Exponents
	Multiplying Fractions
	Multiplication by 50

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Now we will examine an interesting property of i. When we raise it to any positive integer power and simplify, the result is one of only four possibilities: i, -1, -i, or 1.

Look at the powers of i listed in the table.

To simplify a higher power of i, we use this fact: i⁴ = 1.

For example, lets simplify i^10.

i¹⁰

Use the Multiplication Property of Exponents to write i¹⁰ as a product where one factor is a power of i that is a multiple of 4.

= i⁸ · i²

Rewrite i⁸ in terms of i⁴.

Replace i⁴ with 1. Replace i² with -1.

Multiply.

So, i¹⁰ = -1.

= (i⁴)² · i²

= (1)² · i²

= -1

Note:

i¹ = i

i² = -1

i³ = i² · i = (-1) · i = -i

i⁴ = i² · i² = (-1)(-1) = 1

i⁵ = i⁴ · i¹ = 1 · i = i

i⁶ = i⁴ · i² = 1 · (-1) = -1

i⁷ = i⁴ · i³ = 1 · -i = -i

i⁸ = i⁴ · i⁴ = 1 · 1 = 1

i⁹ = i⁴ · i⁴ · i = 1 · 1 · i = i

The pattern repeats: i, -1, -i, 1, i, -1, -i, 1,

We can follow the same process to simplify i²⁷.

Write i²⁷ using a multiple of 4.

Rewrite i²⁴ in terms of i⁴.

Replace i⁴ with 1.

Replace i³ with -i.

Multiply.

i²⁷

= i²⁴ · i³

= (i⁴)⁶ · i³

= 1⁶ · i³

= 1 · (-i)

= -i

So, i²⁷ = -i.

Example 1

Simplify.

a. i³⁵

b. i⁸²

c. i²⁰

Solution

a. To simplify i³⁵, divide 35 by 4. The result is 8 with remainder 3.
i³⁵	= (i⁴)⁸ · i³ = 1⁸ · i³ = 1 · (-i) = -i
b. To simplify i⁸², divide 82 by 4. The result is 20 with remainder 2.
i⁸²	= (i⁴)²⁰ · i² = 1²⁰ · i² = 1 · (-1) = -1
c. To simplify i²⁰, divide 20 by 4. The result is 5 with remainder 0.
i²⁰	= (i⁴)⁵ · i⁰ = 1⁵ · 1 = 1 · 1 = 1