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 Depdendent Variable

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 Dependent Variable

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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:

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 n-th root of a
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
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.

2. If a is negative, then n must be odd for the nth root of a
to be a real number.

e.g.

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 nth-rooting
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 nth-power 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 4th-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.