Before defining the next family of functions, the
exponential functions , we will need to
discuss exponent notation in detail. As we shall see, exponents can be used to
describe
not only powers (such as 5^{2} and 2^{3}), but also roots (such as square roots and
cube
roots). Along the way, we’ll define higher roots and develop a few of their
properties.
More detailed work with roots will then be taken up in the next chapter.
Integer Exponents
Recall that use of a positive integer exponent is simply a shorthand for
repeated multiplication.
For example,
and
In general, b^{n} stands for the quanitity b multiplied by
itself n times. With this definition,
the following Laws of Exponents hold.
Laws of Exponents

The Laws of Exponents are illustrated by the fol lowing
examples .
Example 3.
Note that the second law only makes sense for r > s, since
otherwise the exponent
r − s would be negative or 0. But actually, it turns out that we can create
definitions
for negative exponents and the 0 exponent, and consequently remove this
restriction .
Negative exponents, as well as the 0 exponent, are simply
defined in such a way that
the Laws of Exponents will work for all integer exponents.
• For the 0 exponent, the first law implies that
, and therefore
If
b ≠ 0, we can divide both sides by b to obtain b^{0} = 1 (there is one exception:
0^{0} is
not defined).
• For negative exponents, the second law implies that
provided that b ≠ 0. For example,
, and
Therefore, negative exponents and the 0 exponent are defined as follows:
Definition 4.
and
provided that b ≠ 0. 
Example 5. Compute the exact values of
, and
.
We now have b^{n} defined for all integers n, in such a way
that the Laws of Exponents
hold. It may be surprising to learn that we can likewise define expressions
using rational
exponents, such as , in a consistent manner. Before doing so, however, we’ll
need
to take a detour and define roots.
Roots
Square Roots: Let’s begin by defining the square root of a real number . We’ve
used the square root in many sections in this text, so it should be a familiar
concept.
Nevertheless, in this section we’ll look at square roots in more detail.
Definition 6. Given a real number a , a “square
root of a” is a number x such
that x^{2} = a. 
For example, 3 is a square root of 9 since 3^{2} = 9.
Likewise, −4 is a square root of 16
since (−4)^{2} = 16. In a sense, taking a square root is the “opposite” of
squaring, so the
definition of square root must be intimately connected with the graph of y = x^{2},
the
squaring function. We investigate square roots in more detail by looking for
solutions
of the equation
There are three cases, each depending on the value and
sign of a. In each case, the
graph of the lefthand side of x^{2} = a is the parabola shown in Figures 1(a),
(b), and
(c).
• Case I: a < 0
The graph of the righthand side of x^{2} = a is a horizontal line located a units
below
the xaxis. Hence, the graphs of y = x^{2} and y = a do not intersect and the
equation
x^{2} = a has no real solutions. This case is shown in Figure 1(a). It follows that
a
negative number has no square root.
• Case II: a = 0
The graph of the righthand side of x^{2} = 0 is a horizontal line that coincides
with
the xaxis. The graph of y = x^{2} intersects the graph of y = 0 at one point, at
the vertex of the parabola. Thus, the only solution of x ^{2} = 0 is x = 0, as seen
in
Figure 1(b). The solution is the square root of 0, and is denoted
so it
follows
that
• Case III: a > 0
The graph of the righthand side of x^{2} = a is a horizontal line located a units
above
the xaxis. The graphs of y = x^{2} and y = a have two points of intersection, and
therefore the equation x^{2} = a has two real solutions, as shown in Figure 1(c).
The
solutions of x^{2} = a are Note that we have two notations, one that calls
for the positive solution and a second that calls for the negative solution.
(a) No real solutions. 
(b) One real solution. 
(c) Two real solutions. 
Figure 1. The solutions of x^{2} = a depend upon the sign and
value of a.
Let’s look at some examples.
Example 8. What are the solutions of x^{2} = −5?
The graph of the lefthand side of x^{2} = −5 is the parabola depicted in
Figure
1(a).
The graph of the righthand side of x^{2} = −5 is a horizontal line located 5 units
below
the xaxis. Thus, the graphs do not intersect and the equation x^{2} = −5 has no
real
solutions.
You can also reason as follows. We’re asked to find a solution of x^{2} = −5, so
you
must find a number whose square equals −5. However, whenever you square a real
number, the result is always nonnegative ( zero or positive ). It is not possible
to square
a real number and get −5.
Note that this also means that it is not possible to take the square root of a
negative
number. That is, is not a real number.
Example 9. What are the solutions of x^{2} = 0?
There is only one solution, namely x = 0. Note that this means that
Example 10. What are the solutions of x^{2} = 25?
The graph of the lefthand side of x^{2} = 25 is the parabola depicted in
Figure
1(c).
The graph of the righthand side of x^{2} = 25 is a horizontal line located 25
units above
the xaxis. The graphs will intersect in two points, so the equation x^{2} = 25 has
two
real solutions.
The solutions of x^{2} = 25 are called square roots of 25 and are
written
In this case, we can simplify further and write x = ±5.
It is extremely important to note the symmetry in Figure 1(c) and note that we
have two real solutions, one negative and one positive. Thus, we need two
notations,
one for the positive square root of 25 and one for the negative square root 25.
Note that (5)^{2} = 25, so x = 5 is the positive solution of x^{2} = 25. For the
positive
solution, we use the notation
This is pronounced “the positive square root of 25 is 5.”
On the other hand, note that (−5)^{2} = 25, so x = −5 is the negative solution of
x^{2} = 25. For the negative solution, we use the notation
This is pronounced “the negative square root of 25 is −5.”
This discussion leads to the following detailed summary.
Summary: Square Roots
The solutions of x^{2} = a are called “square roots of a.”
• Case I: a < 0. The equation x^{2} = a has no real solutions.
• Case II: a = 0. The equation x^{2} = a has one real solution, namely x =
0.
Thus,
• Case III: a > 0. The equation x^{2} = a has two real
solutions,The
notation calls for the positive square root of a, that is, the positive
solution
of x^{2} = a. The notation calls for the negative square root of a, that is,
the negative solution of x^{2} = a. 
Cube Roots : Let’s move on to the definition of cube roots.
Definition 11. Given a real number a, a “cube
root of a” is a number x such
that x^{3} = a. 
For example, 2 is a cube root of 8 since 2^{3} = 8. Likewise,
−4 is a cube root of
−64 since (−4)^{3} = −64. Thus, taking the cube root is the “opposite” of cubing,
so the
definition of cube root must be closely connected to the graph of y = x^{3}, the
cubing
function. Therefore, we look for solutions of
Because of the shape of the graph of y = x^{3}, there is only one case to consider.
The
graph of the lefthand side of x^{3} = a is shown in Figure 2. The graph of the righthand
side of x^{3} = a is a horizontal line, located a units above, on, or below the
xaxis,
depending on the sign and value of a. Regardless of the location of the
horizontal line
y = a, there will only be one point of intersection, as shown in Figure 2.
A detailed summary of cube roots follows.
Summary: Cube Roots
The solutions of x^{3} = a are called the “cube roots of a.” Whether a is
negative,
zero, or positive makes no difference. There is exactly one real
solution, namely

Figure 2. The graph of y = x^{3} intersect
the graph of y = a in exactly one
place.
Let’s look at some examples.
Example 13. What are the solutions of x^{3} = 8?
The graph of the lefthand side of x^{3} = 8 is the cubic polynomial shown in
Figure 2.
The graph of the righthand side of x^{3} = 8 is a horizontal line located 8 units
above the
xaxis. The graphs have one point of intersection, so the equation x^{3} = 8 has
exactly
one real solution.
The solutions of x^{3} = 8 are called “cube roots of 8.” As shown from the graph,
there is exactly one real solution of x^{3} = 8, namely
Now since (2)^{3} =
8, it
follows that x = 2 is a real solution of x^{3} = 8. Consequently, the cube root of
8 is 2,
and we write
Note that in the case of cube root, there is no need for the two notations we
saw in the
square root case (one for the positive square root, one for the negative square
root).
This is because there is only one real cube root. Thus, the notation
is
pronounced
“the cube root of 8.”