Free Algebra
Arithmetic Operations with Numerical Fractions
Multiplying a Polynomial by a Monomial
Solving Linear Equation
Solving Linear Equations
Solving Inequalities
Solving Compound Inequalities
Solving Systems of Equations Using Substitution
Simplifying Fractions 3
Factoring quadratics
Special Products
Writing Fractions as Percents
Using Patterns to Multiply Two Binomials
Adding and Subtracting Fractions
Solving Linear Inequalities
Adding Fractions
Solving Systems of Equations -
Exponential Functions
Integer Exponents
Example 6
Dividing Monomials
Multiplication can Increase or Decrease a Number
Graphing Horizontal Lines
Simplification of Expressions Containing only Monomials
Decimal Numbers
Negative Numbers
Subtracting Polynomials
Adding and Subtracting Fractions
Powers of i
Multiplying and Dividing Fractions
Simplifying Complex Fractions
Finding the Coordinates of a Point
Fractions and Decimals
Rational Expressions
Solving Equations by Factoring
Slope of a Line
Percent Introduced
Reducing Rational Expressions to Lowest Terms
The Hyperbola
Standard Form for the Equation of a Line
Multiplication by 75
Solving Quadratic Equations Using the Quadratic Formula
Raising a Product to a Power
Solving Equations with Log Terms on Each Side
Monomial Factors
Solving Inequalities with Fractions and Parentheses
Division Property of Square and Cube Roots
Multiplying Two Numbers Close to but less than 100
Solving Absolute Value Inequalities
Equations of Circles
Percents and Decimals
Integral Exponents
Linear Equations - Positive and Negative Slopes
Multiplying Radicals
Factoring Special Quadratic Polynomials
Simplifying Rational Expressions
Adding and Subtracting Unlike Fractions
Graphuing Linear Inequalities
Linear Functions
Solving Quadratic Equations by Using the Quadratic Formula
Adding and Subtracting Polynomials
Adding and Subtracting Functions
Basic Algebraic Operations and Simplification
Simplifying Complex Fractions
Axis of Symmetry and Vertices
Factoring Polynomials with Four Terms
Evaluation of Simple Formulas
Graphing Systems of Equations
Scientific Notation
Lines and Equations
Horizontal and Vertical Lines
Solving Equations by Factoring
Solving Systems of Linear Inequalities
Adding and Subtracting Rational Expressions with Different Denominators
Adding and Subtracting Fractions
Solving Linear Equations
Simple Trinomials as Products of Binomials
Solving Nonlinear Equations by Factoring
Solving System of Equations
Exponential Functions
Computing the Area of Circles
The Standard Form of a Quadratic Equation
The Discriminant
Dividing Monomials Using the Quotient Rule
Squaring a Difference
Changing the Sign of an Exponent
Adding Fractions
Powers of Radical Expressions
Steps for Solving Linear Equations
Quadratic Expressions Complete Squares
Fractions 1
Properties of Negative Exponents
Factoring Perfect Square Trinomials
Solving Quadratic Equations Using the Square Root Property
Dividing Rational Expressions
Quadratic Equations with Imaginary Solutions
Factoring Trinomials Using Patterns
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Simplifying Expressions Containing only Monomials

Monomials are algebraic expressions consisting of a single term (though this term may contain subexpressions which do contain more than one term). So here, we look at the simplification of simple products, quotients, and powers – in effect, we are reviewing and illustrating the laws of exponents, but using algebraic expressions rather than simple numerical expressions.

Symbolically, we have five distinct rules for combining powers:

Usually simplification involves combining rules (iv) and (v) with one or more of rules (i), (ii), or (iii). When division is involved, rule (ii) brings in something like cancelling common factors between the numerator and the denominator of a fraction. We’ll show a few examples here, but our detailed descriptions of methods for simplifying fractions must wait until a later document in this series.

Remember that multiplication with simple numbers or symbols representing simple numbers is commutative – the order of the factors doesn’t matter:

a · b = b · a


Example 1:

Simplify (3x 2 )(5x 3 )


We can rewrite this expression in detail as

(3x 2 )(5x 3 ) = (3)(x 2 )(5)(x 3 )

the product of four distinct factors. Now

(3)(x 2 )(5)(x 3 ) = (3)(5)(x 2 )(x 3 )

by rearranging the order of the factors, which leaves the result unchanged since the multiplication here is commutative. Now

(3)(x 2 )(5)(x 3 ) = (3 · 5)(x 2 · x 3 ) = 15x 2 + 3 = 15x 5

So, we conclude that

(3x 2 )(5x 3 ) = 15x 5


Example 2:

Simplify (2x 3 )2 (3x 3 )4


Using property (iv)

(2x 3 )2 = (2)2 (x 3 )2 = 4x 3 · 2 = 4x 6


(3x 3 )4 = (3)4 (x 3 )4 = 81x 3 · 4 = 81x 12


(2x 3 )2 (3x 3 )4 = (4x 6 )(81x 12) = (4 · 81)(x 6 · x 12 ) = 324x 6 + 12 = 324x 18


Example 3:

Simplify (-3x 2 )5 (2x 4 )2


Care must be taken with the minus sign here. Again, applying property (iv), we get

(-3x 2 )5 = (-3)5 (x 2 )5 = -243x 10


(2x 4 )2 = (2)2 (x 4 )2 = 4x 8


(-3x 2 )5 (2x 4 )2 = (-243x 10 )(4x 8 ) = -972x 10+8 = -972x 18

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