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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Laws of Exponents

(i) multiplication of two powers:

if c is any number, then

(ii) division of one power by another:

if c is any nonzero number, and m is a larger number than n, then

if c is any nonzero number, and m is a larger number than n, then

(iii) raising a power to a power:

if c is any number, then

(iv) raising a product to a power:

if c and d are any numbers,

(v) raising a quotient or a fraction to a power:

if c is any number, and d is any nonzero number, then

(vi) A Caution!

The five “laws” of exponents summarized in the square boxes above all involve multiplication or division only. As soon as addition or subtraction occurs, the obvious relations are NOT TRUE!

For example,

( 2 + 3 ) 4 = 5 4 = 625


( 2 + 3 ) 4 2 4 + 3 4,

since this latter expression evaluates to 16 + 81 = 97, which is clearly incorrect. Another example is

( 7 - 4 ) 5 = 3 5 = 243


( 7 - 4 ) 5 7 5 - 4 5,

since the latter gives 16807 – 1024 = 15783, which is clearly wrong. In both of these cases, the rules of priority for operations (described just ahead in these topic notes) are followed in the correct forms. It is necessary to evaluate the quantity in brackets first, and then apply the exponent to the result. The exponents here apply to the result of doing whatever operations are shown inside the brackets. This will become a very important rule to remember when the brackets contain symbolic expressions rather than simple numerical expressions.

So, for powers of sums and differences, there is no simple law (such as forms (iv) and (v) above for powers of products and quotients, respectively) that allows us to express them in terms of powers of the original terms in the sums or differences themselves:

All of the properties of powers described in this section will become even more useful when we deal with expressions involving symbols rather than just numbers.

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