Exponents and Their Properties

Think About the Situation
A matryoshka doll, also known as Russian nesting doll,
refers to a set of wooden dolls of decreasing size placed
one inside the other.
In a set of nesting dolls, the smallest doll, Doll 0, is 1 inch
tall. Suppose each doll is twice as tall as the previous doll.
How tall is each of the following dolls?
a. Doll 1
b. Doll 3
c. Doll 8
d. Doll n
The first Russian nested doll set was carved in 1890 by
Vasily Zvyozdochkin.
Investigation 1: Exponents and Their Properties
Perhaps you first used exponents to find areas and volumes. For example, the formula for the area of a square
with side length s is A  s 2 , often read as “s squared.” The volume of a cube with side length s is V  s 3 , often
read as “s cubed.”
There are other applications for the use of exponents. As you have already experienced, exponents can be
used in expressions to represent growth. They can also be used in expressions to represent decay. In solving
problems involving exponential expressions, it helps to know some basic methods for reasoning with and
writing the expressions in equivalent forms.
As you work on problems in this investigation, look for answers to this question:
How can the definition of exponent be used to discover and justify other
properties of exponents that make useful algebraic manipulations possible?
Definition of an exponent
Exponents are used to show repeated multiplication. For example, you can write 3 x 3 x 3 x 3 as 34. You need
to have an understanding of the parts of an exponential expression and their meaning. This will allow you to
make calculations, simplify expressions, and explain and apply the basic rules of exponents.
Let’s look at the parts of the expression 34.
The 4 is the exponent of the expression.
The 3 is the base of the expression.
The expression 34 means that there are “4 factors of 3”.
4 factors of 3
Adapted from Core Plus Course 1 and CME Algebra 1 by K McPherson 1
1. Interpret the meaning of each of the following expressions.
c. 6
b. 105
a. 28
e. 43  52
f. x3 y
d. x 3
g. x
h. (5 x) 2
Products of Powers Work with exponents is often helped by writing products like b x  b y in simpler form or by
breaking a calculation like b z into a product of two smaller numbers.
2. Find values for w, x, and y that will make these equations true statements:
b. 52  54  5 y
a. 210  23  2 y
c. 3  37  3 y
d. 2w  24  27
e. b 4  b 2  b y
f. 9 w  9 x  95
3. Examine the results of your work on Problem 2.
a. What pattern seems to relate task and result in every case?
b. Provide a convincing argument using the definition of exponent to justify your conclusion in part a.
4. Simplify each of the following.
a. ( x 2 y 4 )(3x)
c. (2ab3 )(5a 2 )(ab8 )
b. m5 n 2  m 7 n  m 4 n 4
 3x   5 x3 
d.  2   5 
 y  y 
Power of a Power You know that 8=23 and 64=82, so 64=(23)2. As you work on the next few problems, look for
a pattern suggesting how to write equivalent forms for expressions like (bx)y that involve powers of powers.
5. Find values for x and z that will make these equations true statements.
a. (23 ) 4  2 z
b. (35 ) 2  3z
c. (52 ) x  56
d. (b 2 )5  b z
6. Examine the results of your work on Problem 5.
a. What pattern seems to relate task and result in every case?
b. Provide a convincing argument using the definition of exponent to justify your conclusion in part a.
7. Simplify each of the following.
a. 3( x 3 ) 2
c. (5u 4 ) 2
b. (3 y 5 )2 (2 xy 2 )
 2 y5 
d.  2 
 3x 
3
Adapted from Core Plus Course 1 and CME Algebra 1 by K McPherson 2
Quotients of Powers Since many useful algebraic functions require division of quantities, it is helpful to be
bx
able to simplify expressions involving quotients of powers like y  b  0  .
b
8. Find values for x, y, and z that will make these equations true statements.
109
36
210
2x
z
z
z



10
3
2
 27
a.
b.
c.
d.
5
103
32
23
2
e.
7x
 72
7y
f.
b5
 bz
b3
g.
35
 3z
35
h.
bx
 bz
bx
9. Examine the results of your work on Problem 8.
a. What pattern seems to relate task and result in every case?
b. Provide a convincing argument using the definition of exponent to justify your conclusion in part a.
c. Explain why it is reasonable to define b 0  1 for any base b (b  0) ?
10. Simplify each of the following.
27 m5 n 6
a.
9mn3
c.
14a 5b3
7 a 2b 2
( 2 x 2 y ) 3
b.
9 x2 y 2
d.
x 2n y n
xn y
Summarize the Mathematics
In this investigation, you discovered, tested, and justified several properties of exponents that allow writing of
exponential expressions in equivalent forms.
a. Describe in words the properties for writing exponential expressions in equivalent forms that you
discovered in problems 2-10. These should include Products of Powers, Power of a Power, and
Quotients of Powers.
b. Provide an exponential expression that uses each of the properties at least once and use the definition
of exponents to justify ways to rewrite it.
Check Your Understanding
a. Suppose A  c 2 and B  c 3 . Find two ways to write c 8 in terms of A and B.
(2 x 2 y 3 )3
b. Multiple Choice Question Simplify the following expression
.
4x4 y
1 6 6
1
x y
A.
B. 2xy 5
C. x 2 y 5
D. 2x 2 y 8
2
2
Adapted from Core Plus Course 1 and CME Algebra 1 by K McPherson 3