Laws of Exponents

by
K. D.  Stroyan
University of Iowa

copyright 1997 by Academic Press, Inc. - All rights reserved.

Notebook Overview

This NoteBook reviews the ruels of exponents from high school math.  You need to be sharp on these in order to put formulas into the proper form to apply calculus.

The Rules

For positive numbers  b  and real values of  p  and  q  we have all the following rules of exponents:

Addition of Exponents

(b^p) (b^q)  =  (b^(p + q))

Multiplication of Exponents

((((b^p)))^q) = (b^(p q))

Negative Exponents

(1/(b^p)) = (b^(-p))

Fractional Exponents

(b^(1/p)) =  (b^(1/p))

Combinations of rules

Here are some examples of using sevaral of the rules together:

Fractional Powers & Multiplication of Exponents

(((b^2))^(1/3)) = ((((b^2)))^((1/3)))((((b^2)))^(1/3)) = (b^(2*(1/3))) =  (b^((2/3)))

Negative Powers and Addition of Exponents

((b^5)/(b^2)) =  (b^5)* (b^(-2))(b^5)* (b^(-2)) =  (b^(5 - 2))  =  (b^3)

References to the Text

This NoteBook goes with Chapter 28 of the text on High School Review.

Exercise 28.4.1 Step-by-step

Check your paper and pencil work in Exercise 28.4 by putting the mouse in the next cell, clicking once to get an insert bar and then pressing <Shift>+<Return> or <Enter> on the number pad:

a

(3^2) (3^3)

(3^5)

Notice that exponents do NOT multiply in this case:

(3^6)

b

(5^4) (5^2)

(5^(2 + 4))

c

((((2^4)))^2)

(2^8)

Notice that exponentiation in the other order gives a different result:

(2^((4^2)))

d

((((2^3)))^3)

(2^(3 *3))

(2^((3^3)))

e

(1/(8^4))

(8^(-4))

f - h

Open the basic input palette under File/Palettes/BasicInput and use it to compute the remaining 3 parts of Exercise 28.4.1 both in the form given and in the equivalent form given by laws of exponents.

Exercise 28.4.2

Compare the expressions by entering the computations:

a

\(Clear[x]; 
1\/x\)

(x^(-1))

b

(Sqrt[x])

(x^(1/2))

c

(1/(Sqrt[x]))

(x^(-(1/2)))

d-f

Open the basic input palette under File/Palettes/BasicInput and use it to compute the remaining 3 parts of Exercise 28.4.2 both in the form given and in the equivalent form given by laws of exponents.

Exercise 28.4.3

Compare the expressions by entering the computations:

a

\(Clear[x]; 
\(-\(1\/x\)\)\)

-(x^(-1))

(((-x))^(-1))

b

(x^((2/3)))

(((x^2))^(1/3))

Notice that this is different from:

(((x^3))^(1/2))

c

(x^(-2))

(1/(x^2))

d-f

Open the basic input palette under File/Palettes/BasicInput and use it to compute the remaining 3 parts of Exercise 28.4.2 both in the form given and in the equivalent form given by laws of exponents.