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The FOIL Method
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Multiplication Properties of Exponents
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Absolute Value Function
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Solving Quadratic Equations by Completing the Square
Quotient Rule for Exponents
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Independent, Inconsistent, and Dependent Systems of Equations
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The Vertex of a Parabola
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Test for Factorability for Quadratic Trinomials
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Quadratic Expressions - Completing Squares
Adding and Subtracting Mixed Numbers with Different Denominators
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Multiplying and Dividing Complex Numbers in Polar Form
Power Equations and their Graphs
Solving Linear Systems of Equations by Substitution
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Power of a Product and Power of a Quotient
Factoring Differences of Perfect Squares
Dividing Fractions
Factoring a Polynomial by Finding the GCF
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Multiplication Property of Exponents
Solving Systems of Linear Equations in Three Variables
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Finding the GCF of a Set of Monomials
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Multiplication Property of Exponents

Exponents have several properties. We will use these properties to simplify expressions. In the properties that follow, each variable represents a real number.

Multiplication Property

Property — Multiplication Property of Exponents

English To multiply two exponential expressions with the same base, add their exponents. The base stays the same.

Algebra xm · xn = xm + n (Here, m and n are positive integers.)

Example 54 · 52 = 54 + 2 = 56


Example 1

a. Use the Multiplication Property of Exponents to simplify 23 · 24.

b. Use the definition of exponential notation to justify your answer.


a. The operation is multiplication and the bases are the same. Therefore, add the exponents and use 2 as the base.

23 · 24 = 23 + 4 = 27

b. Rewrite the product to show the factors. Then simplify.


Remember to add the exponents, but leave the bases alone.

That is, 23 · 24 = 23 + 4 = 27 , not 47.

Note the difference between 23 · 24 and 23 + 24.

23 · 24 = 23 + 4 = 27 = 128

23 + 24 = 8 + 16 = 24


Caution — Negative Bases

A negative sign is part of the base only when the negative sign is inside the parentheses that enclose the base.

For example, consider the following cases:

In (-3)2, the base is -3.

(-3)2 = (-3) · (-3) = +9

In -32, the base is 3.

You can think of -32 as the “opposite” of 32.

-32 = -(3 · 3) = -9


Example 2

If possible, use the Multiplication Property of Exponents to simplify each expression:

a. ( 2)2 · ( 2)4

b. 22 · 24

c. 22 · ( 2)4

a. In (-2)2 · (-2)4, the base is -2.

 (-2)2 · (-2)4

= (-2)2 + 4

= (-2)6

= 64

b. In -22 · 24, the base is 2.

We may think of -22 · 24 as the opposite of 22 · 24.

-22 · 24 = -(22) · (24)

= -(22 + 4)

= -(26)

= -64

c. In -22 · (-2)4, the base of the first factor, -22, is 2.

The base of the second factor, (-2)4, is -2.

The bases are not the same, so we cannot use the Multiplication Property of Exponents.

However, we can still evaluate the expression. -22 · (-2)4 = -4 · 16 = -64


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