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Raising an Exponential Expression to a Power
Horizontal Line Test
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Mixed Numbers and Improper Fractions
Solving Quadratic Equations by Completing the Square
Solving Exponential Equations
Adding and Subtracting Polynomials
Factorizing simple expressions
Identifying Prime and Composite Numbers
Solving Linear Systems of Equations by Graphing
Complex Conjugates
Graphing Compound Inequalities
Simplified Form of a Square Root
Solving Quadratic Equations Using the Square Root Property
Multiplication Property of Radicals
Determining if a Function has an Inverse
Scientific Notation
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Exponential Functions
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The Slope of a Line
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Multiplying Polynomials
Slope of a Line
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Percent of Change
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Prime and Composite Numbers
Dividing with Exponents
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Linear Equations and Inequalities in One Variable
Notes on the Difference of 2 Squares
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Factoring Polynomials Completely
Using Slopes to Graph Lines
Fractions, Decimals and Percents
Solving Systems of Equations by Substitution
Quotient Rule for Radicals
Prime Polynomials
Solving Nonlinear Equations by Substitution
Simplifying Radical Expressions Containing One Term
Factoring a Sum or Difference of Two Cubes
Finding the Least Common Denominator of Rational Expressions
Multiplying Rational Expressions
Expansion of a Product of Binomials
Solving Equations
Exponential Growth
Factoring by Grouping
Solving One-Step Equations Using Models
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Adding and Subtracting Polynomials
Rationalizing the Denominator
Rounding Off
The Distributive Property
What is a Quadratic Equation
Laws of Exponents and Multiplying Monomials
The Slope of a Line
Factoring Trinomials by Grouping
Multiplying and Dividing Rational Expressions
Solving Linear Inequalities
Multiplication Property of Exponents
Multiplying and Dividing Fractions 3
Dividing Monomials
Multiplying Polynomials
Adding and Subtracting Functions
Dividing Polynomials
Absolute Value and Distance
Multiplication and Division with Mixed Numbers
Factoring a Polynomial by Finding the GCF
Adding and Subtracting Polynomials
The Rectangular Coordinate System
Polar Form of a Complex Number
Exponents and Order of Operations
Graphing Horizontal and Vertical Lines
Invariants Under Rotation
The Addition Method
Solving Linear Inequalities in One Variable
The Pythagorean Theorem
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Laws of Exponents and Multiplying 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.


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

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".

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.

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 .

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