Systems of Equations
Adding and Subtracting Rational Expressions with Different Denominators
Graphing Linear Equations
Raising an Exponential Expression to a Power
Horizontal Line Test
Quadratic Equations
Mixed Numbers and Improper Fractions
Solving Quadratic Equations by Completing the Square
Solving Exponential Equations
Adding and Subtracting Polynomials
Factorizing simple expressions
Identifying Prime and Composite Numbers
Solving Linear Systems of Equations by Graphing
Complex Conjugates
Graphing Compound Inequalities
Simplified Form of a Square Root
Solving Quadratic Equations Using the Square Root Property
Multiplication Property of Radicals
Determining if a Function has an Inverse
Scientific Notation
Degree of a Polynomial
Factoring Polynomials by Grouping
Solving Linear Systems of Equations
Exponential Functions
Factoring Trinomials by Grouping
The Slope of a Line
Simplifying Complex Fractions That Contain Addition or Subtraction
Solving Absolute Value Equations
Solving Right Triangles
Solving Rational Inequalities with a Sign Graph
Domain and Range of a Function
Multiplying Polynomials
Slope of a Line
Multiplying Rational Expressions
Percent of Change
Equations Involving Fractions or Decimals
Simplifying Expressions Containing only Monomials
Solving Inequalities
Quadratic Equations with Imaginary Solutions
Reducing Fractions to Lowest Terms
Prime and Composite Numbers
Dividing with Exponents
Dividing Rational Expressions
Equivalent Fractions
Graphing Quadratic Functions
Linear Equations and Inequalities in One Variable
Notes on the Difference of 2 Squares
Solving Absolute Value Inequalities
Solving Quadratic Equations
Factoring Polynomials Completely
Using Slopes to Graph Lines
Fractions, Decimals and Percents
Solving Systems of Equations by Substitution
Quotient Rule for Radicals
Prime Polynomials
Solving Nonlinear Equations by Substitution
Simplifying Radical Expressions Containing One Term
Factoring a Sum or Difference of Two Cubes
Finding the Least Common Denominator of Rational Expressions
Multiplying Rational Expressions
Expansion of a Product of Binomials
Solving Equations
Exponential Growth
Factoring by Grouping
Solving One-Step Equations Using Models
Solving Quadratic Equations by Factoring
Adding and Subtracting Polynomials
Rationalizing the Denominator
Rounding Off
The Distributive Property
What is a Quadratic Equation
Laws of Exponents and Multiplying Monomials
The Slope of a Line
Factoring Trinomials by Grouping
Multiplying and Dividing Rational Expressions
Solving Linear Inequalities
Multiplication Property of Exponents
Multiplying and Dividing Fractions 3
Dividing Monomials
Multiplying Polynomials
Adding and Subtracting Functions
Dividing Polynomials
Absolute Value and Distance
Multiplication and Division with Mixed Numbers
Factoring a Polynomial by Finding the GCF
Adding and Subtracting Polynomials
The Rectangular Coordinate System
Polar Form of a Complex Number
Exponents and Order of Operations
Graphing Horizontal and Vertical Lines
Invariants Under Rotation
The Addition Method
Solving Linear Inequalities in One Variable
The Pythagorean Theorem
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Simplifying Expressions Containing only Monomials

We call monomials to those algebraic expressions consisting of a single term (although this term may contain subexpressions which do contain more than one term). In the following excercises we concentrate on the simplification of simple products, quotients, and powers – in fact, we are reviewing and illustrating the laws of exponents, but using algebraic expressions rather than simple numerical expressions.

We could say that "the rules" for combining powers are the following:

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.

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:



Property (iv) can be applied to products of more than two factors. All we need to remember to do is to apply the power outside the brackets to every factor inside the bracket. So

Thus, we get

Rule (i) can be used with a product of more than two factors as well.


Example 2:



Separate all products into factors involving either a number or a power of a single symbol:

Now, rule (i) applies only when the powers being multiplied are powers of the same number or symbol – represented by the symbol ‘a’ in the statement of the rule at the beginning of this document. So, to position the factors in our expression for a potential simplification using rule (i), we rearrange individual factors to get matching symbols in sequence:

Note that in the above work, we used the fact that x = x 1 . Thus, for example

is a product of two factors of x. So, multiplying x 2 by x means the result has an additional factor of x, or three factors of x, hence needs to be written as x 3 . This is what results when we replace x by x 1 and then apply rule (i) for multiplying two powers together. Take care when the expression being simplified has a leading minus sign.


Example 3:



The easiest way to take correct account of the leading minus sign is to keep it attached to the 5 in this case. Then, separating powers of each symbol and rearranging the order of the factors appropriately as we did in the previous example, we get

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