Home
Exponential Functions
Powers
Linera Equations
Simple Trinomials as Products of Binomials
Laws of Exponents and Dividing Monomials
Solving Equations
Multiplying Polynomials
Multiplying and Dividing Rational Expressions
Solving Systems of Linear Inequalities
Mixed-Number Notation
Linear Equations and Inequalities in One Variable
The Quadratic Formula
Fractions and Decimals
Graphing Logarithmic Functions
Multiplication by 111
Fractions
Solving Systems of Equations - Two Lines
Solving Nonlinear Equations by Factoring
Solving Linear Systems of Equations by Elimination
Rationalizing the Denominator
Simplifying Complex Fractions
Factoring Trinomials
Linear Relations and Functions
Polynomials
Axis of Symmetry and Vertices
Equations Quadratic in Form
The Appearance of a Polynomial Equation
Subtracting Reverses
Non-Linear Equations
Exponents and Order of Operations
Factoring Trinomials by Grouping
Factoring Trinomials of the Type ax 2 + bx + c
The Distance Formula
Invariants Under Rotation
Multiplying and Dividing Monomials
Solving a System of Three Linear Equations by Elimination
Multiplication by 25
Powers of i
Solving Quadratic and Polynomial Equations
Slope-intercept Form for the Equation of a Line
Equations of Lines
Square Roots
Integral Exponents
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


 
Try the Free Math Solver or Scroll down to Tutorials!

 

 

 

 

 

 

 

 
 
 
 
 
 
 
 
 

 

 

 
 
 
 
 
 
 
 
 

Please use this form if you would like
to have this math solver on your website,
free of charge.


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, let’s simplify i10. 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