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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
Factoring
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
Algebra
Solving Quadratic Equations Using the Square Root Property
Dividing Rational Expressions
Quadratic Equations with Imaginary Solutions
Factoring Trinomials Using Patterns

Dividing Monomials

Objective Learn the notion of dividing monomials, and the idea of zero and negative numbers as exponents.

It is a good idea that you experiment with computing quotients of powers of 2 in order to see if you can discover the formula for division of powers by yourself.

 

Dividing Powers of the Same Number

To understand division of powers, look at the table that appears below. It shows the results when 64, which equals 2 6 , is divided by various powers of 2.

Do you see any pattern in these numbers? The result is found by subtracting the exponents.

 

Example 1

Divide x 4 by x 2 .

Solution

Expand x 4 into x · x · x · x and x 2 into x · x . Then use these expressions to write a fraction.

Now cancel two of the x ’ s from both numerator and denominator.

It is very important that you understand that “cancellation” just means recognizing that

You should understand that canceling is just shorthand for a process involving the properties of fractions. Any number raised to the power 1 is that number itself.

Quotient of Powers

If we divide a power of a number or variable by another (smaller) power of the same number or variable, the result is the original number raised to the power given by the difference of the two original powers.

Write the quotient and expand the numerator and the denominator into products of x’s.

This shows that the formula for the quotient of powers is valid.

 

Example 2

Simplify

Solution

 

This formula can be used to simplify any quotient of monomials.

 

Example 3

Simplify

Solution

 

Negative Numbers and Zero in the Exponent

We have only defined exponentiation by positive integers, not for zero or for negative numbers. Remind students that when m > n ,

If we let m < n , then we get

On the other hand, , since any number divided by itself is 1. This leads to the following definition.

Zero Exponent

a 0 is defined to be equal to 1.

What would the formula suggest if m < n? For example, let m = 1 and n = 2.

On the other hand,

So, the formula suggests that a -1 should be defined to be . More generally, it suggests that a -n should be defined to be .

Negative Exponents

a -n is defined to be equal to .

At this point, you may be confused by the way the formula was used to generate a definition for negative powers. It was used only as a guide to suggest what a definition should be, but that once we make the definition, we can work with these exponents in exactly the same way that we did with positive exponents. With this definition, these exponents satisfy all the laws of exponents that you should already know. The definition of negative exponents can be used to simplify quotients of monomials.

Example 8

Simplify

Solution

 

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