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Fields Medal Prize Winners (1998)




TUTORIALS:


Solving Quadratic Equations by Using the Quadratic Formula
Addition with Negative Numbers
Solving Linear Systems of Equations by Elimination
Rational Exponents
Solving Quadratic Inequalities
Systems of Equations That Have No Solution or Infinitely Many Solutions
Dividing Polynomials by Monomials and Binomials
Polar Representation of Complex Numbers
Solving Equations with Fractions
Quadratic Expressions Completing Squares
Graphing Linear Inequalities
Square Roots of Negative Complex Numbers
Simplifying Square Roots
The Equation of a Circle
Fractional Exponents
Finding the Least Common Denominator
Simplifying Square Roots That Contain Whole Numbers
Solving Quadratic Equations by Completing the Square
Graphing Exponential Functions
Decimals and Fractions
Adding and Subtracting Fractions
Adding and Subtracting Rational Expressions with Unlike Denominators
Quadratic Equations with Imaginary Solutions
Graphing Solutions of Inequalities
FOIL Multiplying Polynomials
Multiplying and Dividing Monomials
Order and Inequalities
Exponents and Polynomials
Fractions
Variables and Expressions
Multiplying by 14443
Dividing Rational Expressions
Division Property of Radicals
Equations of a Line - Point-Slope Form
Rationalizing the Denominator
Imaginary Solutions to Equations
Multiplying Polynomials
Multiplying Monomials
Adding Fractions
Rationalizing the Denominator
Rational Expressions
Ratios and Proportions
Rationalizing the Denominator
Like Radical Terms
Adding and Subtracting Rational Expressions With Different Denominators
Percents and Fractions
Reducing Fractions to Lowest Terms
Subtracting Mixed Numbers with Renaming
Simplifying Square Roots That Contain Variables
Factors and Prime Numbers
Rules for Integral Exponents
Multiplying Monomials
Graphing an Inverse Function
Factoring Quadratic Expressions
Solving Quadratic Inequalities
Factoring Polynomials
Multiplying Radicals
Simplifying Fractions 1
Graphing Compound Inequalities
Rationalizing the Denominator
Simplifying Products and Quotients Involving Square Roots
Standard Form of a Line
Multiplication by 572
Adding and Subtracting Fractions
Multiplying Polynomials
Factoring Trinomials
Solving Exponential Equations
Solving Equations with Fractions
Roots
Simplifying Complex Fractions
Multiplying and Dividing Fractions
Mathematical Terms
Solving Quadratic Equations by Factoring
Factoring General Polynomials
Adding Rational Expressions with the Same Denominator
The Trigonometric Functions
Solving Nonlinear Equations by Factoring
Solving Systems of Equations
Midpoint of a Line Segment
Complex Numbers
Graphing Systems of Equations
Reducing Rational Expressions
Powers
Rewriting Algebraic Fractions
Exponents
Rationalizing the Denominator
Adding, Subtracting and Multiplying Polynomials
Radical Notation
Solving Radical Equations
Positive Integral Divisors
Solving Rational Equations
Rational Exponents
Mathematical Terms
Rationalizing the Denominator
Subtracting Rational Expressions with the Same Denominator
Axis of Symmetry and Vertex of a Parabola
Simple Partial Fractions
Simplifying Radicals
Powers of Complex Numbers
Fields Medal Prize Winners (1998)

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Multiplying Monomials

Objective Learn what polynomials by knowing how to use the laws of exponents to multiply monomials.

To this point, you surely have worked with linear expressions and equations. Now you need to understand how to do algebra with higher degree expressions, involving higher powers of the variables. The key ideas in this lesson are the laws of exponents, which are used to simplify monomial expressions.

 

Monomials

Let's begin by reviewing exponents. If x is a variable, then x 2 denotes x · x , x 3 denotes x · x · x , and more generally, x n denotes

Definition of Monomials

A monomial is a number, a variable, or a product of numbers and variables.

• Numbers are referred to as constants.

• When there are several occurrences of the same variable, the expression is typically written in exponent form.

Monomials: 5x 2 , 7 xy , -3x 3 , 4y , z 100

Not Monomials:

 

Laws of Exponents and Multiplying Monomials

When multiplying monomials, analyze the product of two powers of the same variable, for instance, x 2 · x 3 . A good way to illustrate this is by multiplying various powers of 2 together.

 

Example 1

Simplify 2 2 · 2 3.

Solution

2 2 · 2 3 = 4 · 8 = 32 = 2 5 2

 

Example 2

Simplify 2 4 · 2 5 .

Solution

2 4 · 2 5 = 16 · 32 = 512 = 2 9

In Examples 1 and 2, the exponent of the result of the multiplication is the sum of the exponents in the two factors. In Example 1, 5 = 2 + 3, and in Example 2, 9 = 4 + 5. Why is this true? Recall that exponents are just shorthand for a repeated product of the same number or variable. So,

In general, when multiplying x m and x n , the result is as follows.

 

Product of Powers

When a power of x is multiplied by another power of x, the result is a power of x whose exponent is the sum of the exponents of the factors.

x m · x n = x m + n

This is true for any number or variable x , and for any whole numbers m and n . Now use this idea to multiply monomials.

 

Example 3

Simplify 3x 5 · 4x 2 .

Solution

3x 5 · 4x 2 = (3 · 4) · ( x 5 · x 2 )
  = 12 x 5 + 2 or 12x 7

Notice that the constants are grouped together and the variables are grouped together.

 

Example 4

Simplif 9y 8 · ( - y 7 ).

Solution

9y 8 · ( - y 7 ) = [9 · ( -1)] · ( y 8 · y 7 ) = -9y 15