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Fields Medal Prize Winners (1998)




TUTORIALS:


Adding and Subtracting Monomials
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 and Dividing Monomials

Objective Learn the algebra of polynomials by using the laws of exponents to multiply and divide monomials.

The main ideas in this lesson are the laws for multiplying and dividing powers. In this lesson, we will deal with monomials that are powers of a single variable.

Powers

Let's begin by reviewing powers and exponents. If a is a variable, then a 2 represents a · a , a 3 represents a · a · a , and more generally, a b represents

a a . . . a . b factors

Remember that a is called the base, b is called the exponent, and a b is called the power. Also, the power a 4 is read "a raised to the fourth power". In general, when we write a b, we say "a raised to the b power".

Laws of Exponents and Multiplying Monomials

When multiplying monomials, we must analyze the product of the two powers of the same base. Consider x 2 · x 3. Let' s analyze this by multiplying various powers of 2 together.

2 2 · 2 3 = 4 · 8 = 32 = 2 5

2 4 · 2 5 = 16 · 32 = 512 = 2 9

In both cases, the exponent of the resulting power is the sum of the exponents in the two factors. For 2 2 · 2 3 , 5 = 2 + 3, and for 2 4 · 2 5 ,9 = 4 + 5. The table shows what happens when a power of 2 is multiplied by "2 to the first power," which is 2. Recall that any number raised to the first power is the number itself.

Notice each power that results. Do you see a pattern? Each resulting power can be found by adding 1 to the exponent of the original power. For example, 2 3 · 2 1 = 2 3 + 1 or 2 4 . Using symbols, we write 2 n · 2 1 = 2 n + 1. The following table shows what happens when a power of 2 is multiplied by "2 to the second power" or 4.

n 2 n 2 n · 2 2 = 2 n · 4
1 2 1 = 2 2 · 4 = 8 or 2 3
2 2 2 = 4 4 · 4 = 16 or 2 4
3 2 3 = 8 8 · 4 = 32 or 2 5
4 2 4 = 16 16 · 4 = 64 or 2 6
5 2 5 = 32 32 · 4 = 128 or 2 7
6 2 6 = 64 64 · 4 = 256 or 2 8

Again, notice each power that results. In this case, each power can be found by adding 2 to the exponent of the original power. For example, 2 3 · 2 2 = 2 3 + 2 or 2 5. Using symbols, we write 2 n · 2 2 = 2 n + 2.

This confirms our earlier observation that when we multiply two powers that have the same base, the exponent of the resulting power is the sum of the exponents in the two factors.

Now is a good time to explore this idea for yourself. Choose your own bases and exponents. Then evaluate both a b · a c and a b + c to verify that they are equal. Why is this true? Remember th at exponents area shorthand that represents a repeated product of the same number or variable. So,

 

In general, when we multiply a b and a c, we'll get

 

Key Idea

When we multiply a power of a times another power of a, the result is a power of a , where the exponent is the sum of the exponents of the two factors. In symbols,

a b · a c = a b + c

This holds true for any number a and positive integers b and c .

 

Laws of Exponents and Dividing Monomials

What happens when we divide powers? Let's analyze this by dividing various powers of 2.

 

Try to make a conjecture about dividing powers that have the same base.

In both cases, the result is the original base raised to the power given by the difference of the two exponents.

For 22 52 , 3 5 2, and for 22 73 , 4 7 3.

The table on the left shows what happens when a power of 2 is divided by 2 1 . The table on the right shows what happens when a power of 2 is divided by 2 2

 

In the table on the left, notice the powers that result. Each resulting power can be found by subtracting 1 from the exponent of the original power. In the table on the right, each resulting power can be found by subtracting 2 from the exponent of the original power. This agrees with our original observation that when we divide two powers with the same base, the exponent of the resulting power is the difference of the exponents of the two dividends.

Consider another case. Let's divide a 4 by a 2. To do this, expand a 4 into a · a · a · a and a 2 into a · a . Next place them into the fraction

 

We can now cancel two a's from both numerator and denominator.

 

Cancellation is a shorthand process involving the properties of fractions. Also, point out that any number raised to the first power is that number itself.

 

After canceling, we find that aa 42 a 4 2 or a 2 .

In general, when we write the quotient and expand it into products of a's, the result is

 

which shows why this fact is valid.

Key Idea

When we divide a power of a by another smaller power of a , the result is a power of a , in which the exponent is the difference of the exponents of the two dividends. In symbols,

aa bc a b c when b c .

This holds true for any nonzero number a and whole numbers b and c.