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THE UNIVERSITY OF AKRON

Mathematics and Computer Science

mptii

Lesson 2: Exponents & Radicals

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IamD S

N ⊆ Z ⊆ Q ⊆ R ⊆ C

a 3 a 4 = a 7 ( ab ) 10 = a 10 b 10

− ( ab − (3 ab − 4))=2 ab − 4

( ab ) 3 ( a − 1 + b − 1 )=( ab ) 2 ( a + b )

( a − b ) 3 = a 3 − 3 a 2 b +3 ab 2 − b 3

2 x 2 − 3 x − 2=(2 x + 1)( x − 2)

2 x +13=0 = ⇒ x = − 26

G= { ( x, y ) | y = f ( x ) }

f ( x )= mx + b

y = sin x

Last Revision Date: 2/2/2000

Lesson 2: Exponents & Radicals

Table of Contents

2. Exponents and Radicals

2.1. Integer Exponents

2.2. The Law of Exponents

• How to Multiply orDivide Two Powers • How to Calcu-

luate a Powerof a Product orQuotient • How to Compute

a Powerof a Power

2.3. Radicals

• Properties of Radicals

2.4. Fractional Exponents

2. Exponents & Radicals

This lesson is devoted to a review of exponents, radicals, and the

infamous “Laws of Exponents.” The student must have the skills to

manipulate exponents without error .

2.1. Integer Exponents

Let a be a number, and n ∈ N be a natural number . The symbol a n

is deﬁned as

a n = a · a · a · a ···a

(1)

n factors

That is, a n is the product of a with itself n times.

Sometimes, negative exponents enterinto the mix. These are deﬁned

a n

a − n =

where n ∈ N and a = 0.

(2)

Needless to say, we deﬁne a 0 = 1, forall a =0.

Section 2: Exponents & Radicals

Thus, the symbol a k is deﬁned forall integers k ∈ Z: forpositive

integers as in equation (1) ,2 3 = (2)(2)(2) = 8; fornegative integers

as in equation (2) ,2 − 3 =1 / 2 3 =1 / 8; and forzero, 2 0 =1.

Terminology. The symbol a k is called a power of a . We say that a k

has a base of a and that k is the exponent of the powerof a .

Numerical calculations oﬀer no challenge to the student (that’s you).

The more interesting case is when there are symbolic quantities in-

volved; however, there is one situation involving numerics (and sym-

bolics) in which some students—I’m not saying you necessarily—have

a weakness. Considerthe following ...

Quiz. Suppose you wanted to square the number − 3, what would be

the correct notational way of writing that?

(a) − 3 2

(b) ( − 3) 2

To eﬀectively manipulate expressions involving symbolics, we must

be the masters of the Laws of Exponents —to be taken up shortly;

just now, however, I want to illustrate how the deﬁnitions of a k are

applied.

Section 2: Exponents & Radicals

Illustration 1. Here are several important illustrations of the tech-

niques revolving about the deﬁnitions given in equations (1) and (2) .

(a) x − 6 =

x 6

y 12 = y − 12

(b)

x 4

x 4 y − 2

z − 5

= x 4 z 5

y 2

w 9

s − 5 t − 3 = w 9 s 5 t 3

Illustration Notes : Look at (a) ﬁrst from left-to-right. Basically, this

says that if we have a negative exponent in the numerator, we can

shove the expression into the denominator by changing the sign of

the exponent. Reading (a) from right-to-left, we see that if we have

an expression in the denominator, we can lift it to the numerator by

changing the sign of the exponent. Similarcomments can be made in

equation (b).

Illustration (c) suggests that if an expression is grouped, then

when we move it to the denominator, we move the whole group.

Examples (d), (e) and (f) demonstrate how equations (1) and

(2) are used in practice. Note that these expressions only involve mul-

tiplication and/or division —most important!

y − 4 = x 4 y 4

(d)

(e)

(f)

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