## Interactive Algebra Review

To assist you with the algebra reviews in Applied Calculus and Finite Mathematics and Applied Calculus

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## 2. Exponents and Radicals Part A: Integer Exponents

Note If you are happy with integer exponents and wish to go on to rational (fractional) exponents, go on to Part B: Radicals and Rational Exponents.

Positive Exponents

If a is any real number and n is any 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 =

The following rules show how to combine such expressions.

Exponent Identities
RuleExample
(a)
 aman = am+n
2322  =   25   =   32
(b)
 aman = am-n   if m > n and a 0
 4342 = 43-2  =   41  =  4
(c)
 (an)m = anm
 (32)2 = 34   =   81
(d)
 (ab)n = anbn
 (4.2)2 = 4222   =   64
(e)
 ab n = anbn
 43 2 = 4232 = 169

 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   7573
=  7 =

Q3   12 2 3
=  12 =

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 = 1an = 1a.a. . . . .a (n times)

If a is any real number other than zero, then we define

a0 = 1.

Examples

 4-3 = 143 = 164
 1,000,0000 = 1
 4x-3 = 4x3
 y-2x3 = 1x3y2

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  x4(x-2)3 = x

Q7  x4y-2(x y2)-2 = x y

Q8  (x-2y)3x4 y-2 3 = x y

Simplify each of the following, and express the answer using no negative exponents. Use formula format, for example x^2/y^4 or x^2*y^4

Q9  x-4y3x-1y-2
=
Q10  x2y2 x -3
=

You should go over Part B in the next tutorial before trying the examples and exercises in Section A.2 of the Algebra Review of Applied Calculus and Finite Mathematics and Applied Calculus

Last Updated: March, 2006