Radical Expressions
9.1 Simplify Radical Expressions
Radical Notation for the nth Root of a
If n is an integer greater than one, then the nth root of a is
the number whose nth power is a. There are two notations
for the nth root of a:
where
n is called the index of the radical
is called the radical symbol
a is called the radicand
is the radical form of the nth root
of a
is the exponential form of the nth
root of a
An expression containing a radical symbol is called a
radical expression. Some examples of radical expressions
are
Consider the Sign of the Radicand a: Positive,
Negative, or Zero
1. If a is positive, then the nth root of a is also a positive
number  specifically the positive number whose nth
power is a.
e.g.
is asking
is asking
2. If a is negative, then n must be odd for the nth
root of a
to be a real number.
e.g.
is asking
is asking
Furthermore, if a is negative and n is odd, then the nth
root of a is also a negative number  specifically the
negative number whose nth power is a.
3. If a is zero, then .
Example 1
1. Evaluate
2. Evaluate
3. Evaluate
4. Evaluate
Square Roots and Cube Roots
1. The second root of a is called the square root of a.
i.e. is read “the square root of a”
2. The third root of a is called the cube root of a.
i.e. is read “the cube root of a”
Definition of
If is a real number, then
where
is the exponential
form of the expression, and
is the radical form of the
expression.
Example 2
Put each expression in radical form.
Example 3
Put each expression in exponential form.
Example 4
Simplify each expression (reduce the index).
Product Property of Radicals
If and
are real numbers, then
In words this tells us the nth root of the product
is the product of nth roots. In terms of the order
of operations, when the only operations are nthrooting
and multiplying, then it does not matter
which operation comes first.
Proof

Condition #1 for a Simplified Radical Expression
A radical expression is not simplified when
the
radicand a has any perfect nthpower factors.
Example 1
Since n = 2, the radicand a can have no perfect square
factors.
Simplify
Simplify
Example 2
Since n = 3, the radicand a can have no perfect cube
factors.
Simplify
Simplify
Example 3
Since n = 4, the radicand a can have no 4^{th}power factors.
Simplify
Simplify
Example 4 Simplify each expression.
Example 5 Simplify each expression.
Condition #2 for a Simplified Radical Expression
The radical expression is not simplified if
m and n
have any common factors. That is, m/n must be in simplest
terms.
Example 6
Simplify each expression.
