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Arithmetic Operations with Numerical Fractions
Multiplying a Polynomial by a Monomial
Solving Linear Equation
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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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Dividing Monomials Using the Quotient Rule

You problably know the definition of division to divide signed numbers. Because the definition of division applies to any division, we restate it here.


Division of Real Numbers

If a, b and c are any numbers with b ≠ 0, then

a ÷ b = c provided that c · b = a.


If a ÷ b = c, we call a the dividend, b the divisor, and c (or a ÷ b) the quotient.

You can find the quotient of two monomials by writing the quotient as a fraction and then reducing the fraction. For example,

You can be sure that x3 is correct by checking that x3 · x2 = x5. You can also divide x2 by x5, but the result is not a monomial:

Note that the exponent 3 can be obtained in either case by subtracting 5 and 2. These examples illustrate the quotient rule for exponents.


Quotient Rule

Suppose a ≠ 0, and m and n are positive integers.

If m ≥ n, then

If n > m, then


Note that if you use the quotient rule to subtract the exponents in x4 ÷ x4, you get the exporession x4 - 4, or x0, which has not been defined yet. Because we must have x4 ÷ x4 = 1 if x ≠ 0, we define the zero power of a nonzero real number to be 1. We do not define the expression 00.


Zero Exponent

For any nonzero real number a, a0 = 1.


Example 1

Using the definition of zero exponent

Simplify each expression. Assume that all variables are nonzero real numbers.

a) 50

b) (3xy)0

c) a0 + b0


a) 50 = 1

b) (3xy)0 = 1

c) a0 + b0 = 1 + 1 = 2

With the definition of zero exponent the quotient rule is valid for all positive integers as stated.


Example 2

Using the quotient rule in dividing monomials

Find each quotient.


Use the definition of division to check that y4 · y5 = y9.

Use the definition of division to check that

Use the definition of division to check that

Use the definition of division to check that x6 · x2y2 = x8y2.


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