Transcript Chapter 7: Rational Exponents and Radicals

§ 7.2
Radical
Expressions and
Functions
Square Roots
The square root of a number is a value that when multiplied by
itself is equal to the original number.
The positive square root is called the principal square root.
Definition of Square Root
If x is a nonnegative real number, then
x is the nonnegative
2
(or principal) square root of x; in other words,
x  x.
Radical
sign
Radicand
 
Index
42
n
x
Radical
expression
When no number for n appears, 2 is the index.
Tobey & Slater, Intermediate Algebra, 5e - Slide #2
Higher Order Roots
Definition of Higher Order Roots
1. If x is a nonnegative real number, then n x is a nonnegative
nth root and has the property that
n
n
x  x.
2. If x is a negative real number, then
 
a.)
b.)
 x
 x
n
n
n
n
 x, when n is an odd integer.
is not a real number, when n is an even integer.
Example: Find the root of
3
3
1000.
“Cube root”
1000  3 (10)3  10
Tobey & Slater, Intermediate Algebra, 5e - Slide #3
Square Root Functions
The square root function f(x) has a domain of all real numbers x
that are greater than or equal to 0.
Example: Find f(3) for the function f ( x)  3x  5, and find the
domain.
f (-3) = -3(-3) - 5
 95
 4
2
To find the domain, we know that 3x – 5 must be nonnegative.
3x  5  0
3x  5
x
The domain is all real numbers
5
3
x where x   5 .
3
Tobey & Slater, Intermediate Algebra, 5e - Slide #4
Expressions with Rational Exponents
If n is a positive integer and x is a nonnegative real number, then
x1/ n  n x.
Example: Change w3/4 to radical form.
w3/4  4 w3
Example: Change (3abc)2/5 to radical form.
3abc
2/

5
3abc
2
 5 9a 2b2c2
Tobey & Slater, Intermediate Algebra, 5e - Slide #5
Evaluating Higher Order Radicals
For all real numbers x (including negative real numbers), then
n
x 2  x when n is an even positve integer,
n
x 2  x when n is an odd positve integer.
Example: Simplify.
a.)
4
x 4 y 20
4
x 4 y 20  xy 5
b.)
3
-27a 3
3
27a 6  3a 2
Tobey & Slater, Intermediate Algebra, 5e - Slide #6