Sect R.2 Rules of Exponents Inquiry Activity
Definition. A power is an expression involving exponents, which indicates a repeated
multiplication.
Definition. A base is the number in a power which indicates the factor that is
repeatedly multiplied.
Definition. An exponent is the number in a power which indicates the number of times
the base is a factor.
1. Consider the expression, 5 4 .
(a) What multiplication problem is indicated by this expression?
(b) What is the base?
(c) What is the power (refer to the definition above)?
2. Consider the expressions x  x and x  x  x .
(a) Find the product of x  x times x  x  x write that result as a multiplication
problem, as well as a power.
(b) Rewrite each of the original expressions, that is both x  x and x  x  x , as a
power.
(c) Based on your responses to parts (a) & (b), what can be done to simplify the
product of two powers?
3. Consider the expression,
x
. To what does this expression simplify? Why?
x
xxxxx
. To what multiplication problem does this
xxx
expression simplify? Determine this without powers.
4. Consider the expression,
5. Reconsider the expression in #4.
(a) Rewrite both the numerator and denominator as powers of x.
(b) Based on your response to #4, what can be done to simplify a rational
expression involving the division of powers (like that in part (a))?
6. Consider the expression y 3 .
(a) What multiplication problem is indicated by this expression?
(b) Suppose y  a5 . Using the substitution principal, rewrite the multiplication
problem from part (b) in terms of a. Then, simplify as a single power of a.
(c) What can be done to simplify an expression like (x 5 ) 3 ?
Basic Properties of Exponents
Problems #2 – 6 summarize and justify the three basic properties of exponents.
Complete the following generalizations of those properties.
Product Rule: a m a n  _______________
 
Power of a Power Rule: a m
n
Quotient Rule:
am
 ________________
an
 _______________
x
. What exponent is understood to be on each x? Using
x
the quotient rule simplify this rational expression to a single power of x.
7. Reconsider the expression
8. The zero exponent rule is justified by #7, which states that a 0  __________ ,
unless
.
9. Write a relatively simple rational expression involving two powers of x that would,
according to the quotient rule, simplify to x 3 .
10. Using the rational expression that you created in #9, explain why x 3 . 
1
.
x3
11. The negative exponent rule is summarized by #10, and states a m  __________ .
12. Simplify completely.
1
x 8
The last two important properties about exponents are the power of a product
m
property which states that ab  a m b m and the power of a quotient property which
m
am
a
states that    m .
b
b
Now, take a close look at some problems that involve two or more of the Power Rules.
Remember to follow order of operations.
6a 3 b 2
.
4a 2 b 7
(a) What can be done to simplify the coefficients in the numerator and
denominator?
13. Consider the expression
(b) What can be done to simplify the powers of a? What can be done to simplify
the powers of b?
(d) Now, simplify the expression
6a 3 b 2
.
4a 2 b 7
Practice. Simplify each of the following expressions, using the power rules.
15.
 
3x 5
x3
17. 3  3
n

16. 6 y 2 2 y 0
 a 2
18.   2
b
2n
19.  2 x
 4x 
2 3
3 1
2
 b 
 
 a 
 7a 5 b 3 
20.   2 
 3a b 
3
2