MA099 - Spring2013
7.1 Radical Expressions, Functions, and Models
Keynote

Square Roots and Square Root Functions

Cube Roots

Odd and Even nth Roots

Radical Functions
Definition

The number 𝑐 is a square root of 𝑎 if 𝑐 2 = 𝑎

The principal square root of a nonnegative number is its nonnegative square root.

The symbol √
is called a radical sign
and indicates the principal square root of the number over which it appears.

√a is read as “the square root of 𝑎,” “root 𝑎,” or “radical 𝑎.”

The expression under the radical sign is called the radicand.
Example Find two square roots of 16.
Example Simplify each of the following
b) − √
a) √81
16
81
Expressions of the Form √𝒂𝟐
It is tempting to write √𝒂𝟐 = 𝒂 but the next example shows that, as a rule, this is untrue.
Example
a)
82  64  8
b)
(8)2  64  8
( (8)2  8)
Simplifying √𝒂𝟐
For any real number 𝑎,
√𝒂𝟐 = |𝐚|
Example Simplify each expression. Assume that the variable can represent any real number.
a)
( y  3) 2
b)
m12
c)
x10
Definition The number 𝑐 is a cube root of 𝑎 if 𝑐 3 = 𝑎
Example Simplify 3 27 x3 .
Odd and Even 𝒏𝒕𝒉 Roots

The fourth root of a number 𝒂 is the number 𝑐 for which 𝑐 4 = 𝑎.

We write √𝑎 for the 𝒏th root.
𝑛
The number 𝑛 is called the index (plural, indices).
When the index is 2, we do not write it.
Example Find each of the following
a) 5 243
a) 4 81
b) 5 243
b) 4 81
c)
11 11
m
c)
4
16m 4
Summary: Simplifying nth Roots
Radical Functions
A radical function is a function that can be described by a radical expression.
Example
Find the domain of the function given by the equation. Check by graphing the function. Then, from
the graph, estimate the range of the function.
𝑓(𝑥) = √𝑥 − 4