College Algebra 9th txtbk

R-2 Exponents and Radicals
Z Integer Exponents
Z Scientific Notation
Z Roots of Real Numbers
Z Rational Exponents and Radicals
Z Simplifying Radicals
The French philosopher/mathematician René Descartes (1596–1650) is generally credited
with the introduction of the very useful exponent notation “x n .” This notation as well as
other improvements in algebra may be found in his Geometry, published in 1637.
If n is a natural number, x n denotes the product of n factors, each equal to x. In this section,
the meaning of x n will be expanded to allow the exponent n to be any rational number.
Each of the following expressions will then represent a unique real number:
SECTION R–2 Exponents and Radicals 11
7 5 5 4 3.14 0 6 1 2
14 5 3
Z Integer Exponents
If a is a real number, then
a 6 a a a a a a
6 factors of a
In the expression a 6 , 6 is called an exponent and a is called the base.
Recall that a 1 , for a 0, denotes the multiplicative inverse of a (that is, 1a). To generalize
exponent notation to include negative integer exponents and 0, we define a 6 to be
the multiplicative inverse of a 6 , and we define a 0 to be 1.
Z DEFINITION 1 a n , n an Integer and a a Real Number
For n a positive integer and a a real number:
a n a a . . . a
a n 1 a n
a 0 1
n factors of a
(a 0)
(a 0)
EXAMPLE 1 Using the Definition of Integer Exponents
Write parts (A) and (B) in decimal form and parts (C) and (D) using positive exponents.
Assume all variables represent nonzero real numbers.
(A)
(u 3 v 2 ) 0
(B)
10 3
(C)
x 8
(D) x3
y 5