9.1 Simplify Radical Expressions
Radical Notation for the n-th 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:
n is called the index of the radical
is called the radical symbol
a is called the radicand
is the radical form of the n-th root
is the exponential form of the n-th
root of a
An expression containing a radical symbol is called a
radical expression. Some examples of radical expressions
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.
2. If a is negative, then n must be odd for the nth
root of a
to be a real number.
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 .
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”
If is a real number, then
is the exponential
form of the expression, and
is the radical form of the
Put each expression in radical form.
Put each expression in exponential form.
Simplify each expression (reduce the index).
Product Property of Radicals
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 nth-rooting
and multiplying, then it does not matter
which operation comes first.
Condition #1 for a Simplified Radical Expression
A radical expression is not simplified when
radicand a has any perfect nth-power factors.
Since n = 2, the radicand a can have no perfect square
Since n = 3, the radicand a can have no perfect cube
Since n = 4, the radicand a can have no 4th-power factors.
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
Simplify each expression.