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Exponents and Roots

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 52 and 23), 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, bn 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 b0 = 1 (there is one exception: 00 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 bn 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 x2 = a.

For example, 3 is a square root of 9 since 32 = 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 = x2, 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 left-hand side of x2 = a is the parabola shown in Figures 1(a), (b), and
(c).

• Case I: a < 0

The graph of the right-hand side of x2 = a is a horizontal line located a units below
the x-axis. Hence, the graphs of y = x2 and y = a do not intersect and the equation
x2 = 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 right-hand side of x2 = 0 is a horizontal line that coincides with
the x-axis. The graph of y = x2 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 right-hand side of x2 = a is a horizontal line located a units above
the x-axis. The graphs of y = x2 and y = a have two points of intersection, and
therefore the equation x2 = a has two real solutions, as shown in Figure 1(c). The
solutions of x2 = 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 x2 = a depend upon the sign and value of a.

Let’s look at some examples.

Example 8. What are the solutions of x2 = −5?

The graph of the left-hand side of x2 = −5 is the parabola depicted in Figure 1(a).
The graph of the right-hand side of x2 = −5 is a horizontal line located 5 units below
the x-axis. Thus, the graphs do not intersect and the equation x2 = −5 has no real
solutions.

You can also reason as follows. We’re asked to find a solution of x2 = −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 x2 = 0?

There is only one solution, namely x = 0. Note that this means that

Example 10. What are the solutions of x2 = 25?

The graph of the left-hand side of x2 = 25 is the parabola depicted in Figure 1(c).
The graph of the right-hand side of x2 = 25 is a horizontal line located 25 units above
the x-axis. The graphs will intersect in two points, so the equation x2 = 25 has two
real solutions.

The solutions of x2 = 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 x2 = 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
x2 = 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 x2 = a are called “square roots of a.”

• Case I: a < 0. The equation x2 = a has no real solutions.

• Case II: a = 0. The equation x2 = a has one real solution, namely x = 0.
Thus,

• Case III: a > 0. The equation x2 = a has two real solutions,The
notation calls for the positive square root of a, that is, the positive solution
of x2 = a. The notation calls for the negative square root of a, that is,
the negative solution of x2 = 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 x3 = a.

For example, 2 is a cube root of 8 since 23 = 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 = x3, the cubing
function. Therefore, we look for solutions of



Because of the shape of the graph of y = x3, there is only one case to consider. The
graph of the left-hand side of x3 = a is shown in Figure 2. The graph of the right-hand
side of x3 = a is a horizontal line, located a units above, on, or below the x-axis,
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 x3 = 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 = x3 intersect
the graph of y = a in exactly one
place.

Let’s look at some examples.

Example 13. What are the solutions of x3 = 8?

The graph of the left-hand side of x3 = 8 is the cubic polynomial shown in Figure 2.
The graph of the right-hand side of x3 = 8 is a horizontal line located 8 units above the
x-axis. The graphs have one point of intersection, so the equation x3 = 8 has exactly
one real solution.

The solutions of x3 = 8 are called “cube roots of 8.” As shown from the graph,
there is exactly one real solution of x3 = 8, namely Now since (2)3 = 8, it
follows that x = 2 is a real solution of x3 = 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.”

 

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