Game Development Reference
In-Depth Information
Radicals and Roots
In a previous section, you examined the fundamentals of using exponents. You
can also express exponents as fractions, and when you do so you enter the realm
of roots and radicals. To explore a root, begin with the notion that to discover the
power of a number, you begin by multiplying the number by itself, as the dis-
cussion in Chapter 2 detailed:
a a ¼ a
2
3
3
¼
9
The square of 3 is 9
:
a a a ¼ a
3
3
3
3
¼
27
The cube of 3 is 27
:
If you raise 3 by the power of 2, you obtain its square. If you raise 3 by the power
of 3, you obtain its cube. Suppose now that you begin with 9 and 27. You ask the
following questions:
n
Given 9, what number can you multiply by itself to arrive at 9? What is the
square root of 9?
n
Given 27, what number can you raise by a power of 3, or multiply by itself
three times to arrive at 27? What is the cube root of 27?
To represent a root, you use one of two options. First, you can employ a fractional
exponent. The denominator indicates the degree or value of the root. The
numerator indicates the power to which you are raising the number. A numerator
of 1 allows you to indicate any simple root you choose. As a second option, you
can translate the exponent using a radical sign (
). If you use the radical sign
alone, by convention it indicates the square root of the number it designates. Here
are examples of exponential and radical approaches to expressing roots:
9
2
H
p
¼
3
¼
3
p
8
8
3
¼
¼
2
You can work with any number of fractional forms of exponents, and these allow
you to express powers and roots simultaneously. Consider the following expression:
p
3
5
3
2
2
¼
As Figure 3.4 illustrates, positive exponents render radical expressions in which
the numerator designates the power and the denominator then serves as the root.
Figure 3.5 illustrates a further extension of such activity. When you work with a
negative exponent, the result, as with other exponents, renders the multiplicative
inverse of the entire radical expression.
