Exponents and Radicals
If a is any real number and n is a positive integer, then the nth power of a is
The number a is called the base and n is called the exponent.
Zero and Negative Exponents
If a ≠ 0 is any real number and n is a positive integer, then
Laws of Exponents
Example 1. Evaluate the expressions
Example 2. Simplify the expressions
||law 4 and 3
||factor 6 and use law 1
||definition of negative exponents
means b2 = a and b ≥ 0.
, since 32 = 9 and 3 ≥0.
Definition of nth Root
If n > 1 is a positive integer, then the principal nth root of a,
denoted by is
(1) 0 if a = 0.
(2) the positive number b such that bn = a, if a is positive.
(3) (a) the negative number b such that bn = a, if n is odd and a is negative.
(b) not a real number if n is even and a is positive.
If n is even, we must have a ≥ 0 and b ≥ 0. Complex
numbers are needed to define if
a < 0 and n is an even positive integer, because when n is even for all real
numbers b, we
have bn ≥ 0.
If n = 2 we write instead of
Example. since 33 = 81 and 3 ≥ 0.
Example. since (-2)3 = -8.
Note that are not defined as real numbers.
Properties of nth Roots
Note that we have . For
Definition. If m and n are integers and n > 1, and if a is a real
number such that
exists, we define
Note that by the definition
Remark. The laws of exponents hold for rational exponents also.
Example Evaluate the expressions
Example Simplify the expressions
Example Simplify the expression and
eliminate any negative exponent(s).
Rationalizing the Denominator
Rationalizing the denominator is to eliminate the radical in a
denominator by multiplying
both numerator and denominator by an appropriate expression.
In general, if the denominator is of the form n
with m < n, then multiplying the
numerator and denominator by
Example Rationalize the denominator
More Examples. Simplify the expressions and
eliminate any negative exponent(s)