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Adding Fractions
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Multiplication by 50
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Simple Trinomials as Products of Binomials
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Multiplication by 25
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Solving Quadratic Equations by Completing the Square
Quotient Rule for Exponents
Simplifying Square Roots
Multiplying and Dividing Rational Expressions
Independent, Inconsistent, and Dependent Systems of Equations
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Zero Power Property of Exponents
The Vertex of a Parabola
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Test for Factorability for Quadratic Trinomials
Trinomial Squares
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Solving Linear Equations Containing Fractions
Multiplying by 125
Exponent Properties
Multiplying Fractions
Adding and Subtracting Rational Expressions With the Same Denominator
Quadratic Expressions - Completing Squares
Adding and Subtracting Mixed Numbers with Different Denominators
Solving a Formula for a Given Variable
Factoring Trinomials
Multiplying and Dividing Fractions
Multiplying and Dividing Complex Numbers in Polar Form
Power Equations and their Graphs
Solving Linear Systems of Equations by Substitution
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Laws of Exponents
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Systems of Linear Equations
Properties of Rational Exponents
Power of a Product and Power of a Quotient
Factoring Differences of Perfect Squares
Dividing Fractions
Factoring a Polynomial by Finding the GCF
Graphing Linear Equations
Steps in Factoring
Multiplication Property of Exponents
Solving Systems of Linear Equations in Three Variables
Solving Exponential Equations
Finding the GCF of a Set of Monomials
 

Power of a Product and Power of a Quotient

Power of a Product

Consider an example of raising a monomial to a power. We will use known rules to rewrite the expression.

(2x)3 = 2x · 2x · 2x Definition of exponent 3
  = 2 · 2 · 2 · x · x · x Commutative and associative properties
  = 23x3 Definition of exponents

Note that the power 3 is applied to each factor of the product. This example illustrates the power of a product rule.

 

Power of a Product Rule

If a and b are real numbers and n is a positive integer, then (ab)n = an bn.

 

Example 1

Using the power of a product rule

Simplify. Assume that the variables are nonzero.

a) (xy3)5

b) (-3m)3

c) (2x3y2z7)3

Solution

a) (xy3)5 = x5(y3)5 Power of a product rule
  = x5y15 Power rule
b) (-3m)3 = (-3)3m3 Power of a product rule
  = 27m3 (-3)(-3)(-3) = -27

c) (2x3y2z7)3 = 23(x)3(y2)3(z7)3 = 8x9y6z21

 

Power of a Quotient

Raising a quotient to a power is similar to raising a product to a power:

Definition of exponent 3

Definition of multiplication of fractions

Definition of exponents

The power is applied to both the numerator and denominator. This example illustrates the power of a quotient rule.

 

Power of a Quotient Rule

If a and b are real numbers, b ≠ 0, and n is a positive integer, then

Example 2

Using the power of a quotient rule

Simplify. Assume that the variables are nonzero.

Solution

a) Power of a quotient rule

 

(5x3)2 = 5x(x3)2 = 25x6

b) Power of a quotient and power of a product rules

 

Simplify

c) Use the quotient rule to simplify the expression inside the parentheses before using the power of a quotient rule.

 

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