Elimination Using Multiplication
Prime Factors
Equations Involving Rational Exponents
Working with Percentages and Proportions
Rational Expressions
Interval Notation and Graphs
Simplifying Complex Fractions
Dividing Whole Numbers with Long Division
Solving Compound Linear Inequalities
Raising a Quotient to a Power
Solving Rational Equations
Solving Inequalities
Adding with Negative Numbers
Quadratic Inequalities
Dividing Monomials
Using the Discriminant in Factoring
Solving Equations by Factoring
Subtracting Polynomials
Cube Root
The Quadratic Formula
Multiply by the Reciprocal
Relating Equations and Graphs for Quadratic Functions
Multiplying a Polynomial by a Monomial
Calculating Percentages
Solving Systems of Equations using Substitution
Comparing Fractions
Solving Equations Containing Rational Expressions
Factoring Polynomials
Negative Rational Exponents
Roots and Radicals
Intercepts Given Ordered Pairs and Lines
Factoring Polynomials
Solving Linear Inequalities
Mixed Expressions and Complex Fractions
Solving Equations by Multiplying or Dividing
The Addition Method
Finding the Equation of an Inverse Function
Solving Compound Linear Inequalities
Multiplying and Dividing With Square Roots
Exponents and Their Properties
Equations as Functions
Factoring Trinomials
Solving Quadratic Equations by Completing the Square
Dividing by Decimals
Lines and Equations
Simplifying Complex Fractions
Graphing Solution Sets for Inequalities
Standard Form for the Equation of a Line
Checking Division with Multiplication
Elimination Using Addition and Subtraction
Complex Fractions
Multiplication Property of Equality
Solving Proportions Using Cross Multiplication
Product and Quotient of Functions
Quadratic Functions
Solving Compound Inequalities
Operating with Complex Numbers
Equivalent Fractions
Changing Improper Fractions to Mixed Numbers
Multiplying by a Monomial
Solving Linear Equations and Inequalities Graphically
Dividing Polynomials by Monomials
Multiplying Cube Roots
Operations with Monomials
Properties of Exponents
Mixed Numbers and Improper Fractions
Equations Quadratic in Form
Simplifying Square Roots That Contain Whole Numbers
Dividing a Polynomial by a Monomial
Writing Numbers in Scientific Notation
Solutions to Linear Equations in Two Variables
Solving Linear Inequalities
Multiplying Two Mixed Numbers with the Same Fraction
Special Fractions
Solving a Quadratic Inequality
Parent and Family Graphs
Solving Equations with a Fractional Exponent
Evaluating Trigonometric Functions
Solving Equations Involving Rational Expressions
Laws of Exponents
Multiplying Polynomials
Vertical Line Test
Solving Inequalities with Fractions and Parentheses
Multiplying Polynomials
Solving Quadratic and Polynomial Equations
Extraneous Solutions
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Operations with Monomials

A monomial is an algebraic expression that is a constant or a product of a constant and one or more variables with whole number exponents.


Basic Rules for Exponents

1. Extending the Multiplying of “powers with the same base”: a m · a n · a p =a m + n + p

Add exponents by the rules of adding signed numbers where m, n, p { integers}

Example 1:

a) z 5 · z 2 · z 4 = z 5+2+4 = z 11

b) x 2 · x 3 · x 4 · x 5 = x 2+3+4+5 = x 14


2. Powering of powers:

For base a and exponents m, n { natural numbers }

(a m) n = a m·n

Example 2:

a) (x 4 )3 = x 4·3 = x 12

b) (y 2) 5 = y2·5 = y 10


3. Powering of products with different bases a and b, and exponent n { natural numbers}

(a·b) n = a n · b n

NOTE: The exponent affects only the number that it immediately touches so you must always put negative numbers in parentheses ( ).

Example 3:

1. (x 5 y 7)3 = (x 5)3·(y 7)3 = x5·3·y7·3 = x15 y21 Apply properties and multiply exponents.

2. (-2 · x2y3)4 = (-2)4 ·(x2)4·(y3)4 = 16 ·(x2·4)·(y3·4) = 16 · x8y12 Apply properties, multiply exponents.

Definition of Negative Exponent Note: a -n × a n = 1 They are reciprocals.

The sign in the exponent does not affect the sign of the number in the base.

Example 4:

a) Since -2 is in the denominator place positive in the numerator:

b) x -8  Since -8 is in the numerator place positive in the denominator:

Definition of Zero Exponent a 0 = 1 Note: The result of raising any base to the 0 exponent will always be 1. (a ≠ 0)

1. (57x9 y32 z −25 )0 = 1


4. Division of powers with positive exponents:

Subtract the smaller exponent from the larger exponent and put the resulting “power ” in the “place” of the larger “power”.

Put a 1 in the numerator if the larger “power” is in the denominator.


Example 5: (Working with positive exponents):

1. [7 > 3 result goes in the numerator.]

2. [9 > 5 result goes in the denominator.]

3. [4 = 4 Exponents the same result is always 1.]


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