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This topic can calso be found in Section 0.2 of Applied Calculus and Finite Mathematics

Positive Exponents

If a is a real number and n is a positive integer, then by an we mean the quantity

a.a. . . . a   (n times).

The number a is called the base and the number n is called the exponent.

Thus, a1 = a,   a2 = a.a,   a5 = a.a.a.a.a

Here are some examples with actual numbers:

 32 = 9 Base 3 exponent 2 23 = 8 034 = 0 0 to any positive power is 0 (-1)5 = -1 Q1 25 = Q2 (-2)4 = Q3 -24 = Q4 -4^3 = Q5 (-4)^2 = Q6 -(-1)^2 =

The following rules show how to combine such expressions.

Exponent Identities
RuleExample
(a)
 aman = am+n
2322  =   25   =   32
(b)
 am an = am-n   if m > n and a ≠ 0
 43 42 = 43-2  =   41  =  4
(c)
 (an)m = anm
 (32)2 = 34   =   81
(d)
 (ab)n = anbn
 (4.2)2 = 4222   =   64
(e)
 a b n = an bn
 4 3 2 = 42 32 = 16 9

 Caution In identities (a) and (b), the bases of the expressions must be the same. For example, rule (a) gives 3234 = 36, but does not apply to 3242. People sometimes invent their own identities, such as am + an = am+n, which is wrong! (If you don't believe this, try it out with a = m = n = 1.) If you wind up with something like 23 + 24, you are stuck with it -- there are no identities around to simplify it further.

Fill in the missing exponents and other numbers and press "Check.". (Raised boxes are exponents.)

Q1   (-2)4(-2)2
=  (-2) =

Q2   75 73
=  7 =

Q3   1 2 2 3
=  1 2 =

Q4   (xy3)2
=  x y

Q5   (4x2y)3
=  x y

Q6   x9 (x2)3
=  x

Q7   x4y5 (x y2)2
=  x y

Negative and Zero Exponents

It turns out to be very useful to allow ourselves to use exponents that are not positive integers. These are dealt with by the following definition.

Negative and Zero Exponents

If a is any real number other than zero and n is any positive integer, then we define

 a-n = 1 an = 1 a.a. . . . .a (n times)
and a0 = 1.

Examples

 4-3 = 1 43 = 1 64
 1,000,0000 = 1
 4 x-3 = 4x3
 y-2 x3 = 1 x3y2

Here are some for you to try.

Q1  10-5 =

Q2  (-2)-4 =

Q3  (-1)-5 =

Q4  (-3)0 =

Q5
x-4 x2

=

1 x

Q6  3 5 x2 = 5

Q7  x4 (x-2)3 = x

Q8  x4y-2 (x y2)-2 = x y

Q9  (x-2y)3 x4 y-2 3 = x y

Rational and Exponential Form
 An expression of rational form is an expression written as a ratio: axm bxn where m and n ≥ 0.

Examples: Expressions in Rational Form:
 2.1x2 2x4 , 4 7x4 , -3x4 2.3 , 22 7
 but 4x-4 7 is not in rational form because the exponent of x is negative.

 An expression of exponential form is an expression written as axn where n is any exponent (possibly negative or zero).

Examples: Expressions in Exponential Form:
 4.1x-3 , 2 3 x2 , 2 5.1 x-4 , 22 7
 but 4 7x-4 is not in exponential form because of the x is in the denominator.

Converting between Rational and Exponential Form

We can use the exponent identities to convert between the two forms just described:

Rational FormExponential Form
 4 7x4
 4 7 x-4
 2.1x2 2x4
 1.05x-2
 4x4 7
 4 7 x4
 3 2x5
 2 7x-1
 3.2x-6
 -4 3 x-1

Simplify each of the following, and express the answer using no negative exponents.

Q1  x-4y3 x-1y-2
=
Q2  x2 y2 x -3
=

You should go over Part B in the next tutorial before trying the examples and exercises in Section 0.2 of the Algebra Review of Applied Calculus and Finite Mathematics and Applied Calculus

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Last Updated: December, 2007