Exponents and Radicals
Exponential Notation
If a is any real number and n is a positive integer, then the nth power of a is
![](./articles_imgs/5027/expone14.jpg)
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
![](./articles_imgs/5027/expone15.jpg)
Laws of Exponents
![](./articles_imgs/5027/expone16.jpg)
Example 1. Evaluate the expressions
![](./articles_imgs/5027/expone17.jpg)
Example 2. Simplify the expressions
![](./articles_imgs/5027/expone18.jpg)
Solution.
![](./articles_imgs/5027/expone19.jpg) |
law 4 and 3 |
![](./articles_imgs/5027/expone20.jpg) |
rearrange factors |
![](./articles_imgs/5027/expone21.jpg) |
factor 6 and use law 1 |
![](./articles_imgs/5027/expone22.jpg) |
|
![](./articles_imgs/5027/expone23.gif) |
definition of negative exponents |
Radicals
means b2 = a and b ≥ 0.
Example.
, 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
![](./articles_imgs/5027/expone33.jpg)
Note that we have
. For
example,![](./articles_imgs/5027/expone35.jpg)
Rational Exponents
Definition. If m and n are integers and n > 1, and if a is a real
number such that ![](./articles_imgs/5027/expone36.gif)
exists, we define
or equivalently
![](./articles_imgs/5027/expone38.jpg)
Note that by the definition ![](./articles_imgs/5027/expone39.gif)
Remark. The laws of exponents hold for rational exponents also.
Example Evaluate the expressions
![](./articles_imgs/5027/expone40.jpg)
Example Simplify the expressions
![](./articles_imgs/5027/expone41.jpg)
Example Simplify the expression and
eliminate any negative exponent(s).![](./articles_imgs/5027/expone42.jpg)
Rationalizing the Denominator
Rationalizing the denominator is to eliminate the radical in a
denominator by multiplying
both numerator and denominator by an appropriate expression.
Example. ![](./articles_imgs/5027/expone43.jpg)
In general, if the denominator is of the form n
with m < n, then multiplying the
numerator and denominator by ![](./articles_imgs/5027/expone45.jpg)
Example Rationalize the denominator
![](./articles_imgs/5027/expone46.jpg)
More Examples. Simplify the expressions and
eliminate any negative exponent(s)
![](./articles_imgs/5027/expone47.jpg)