Radical Expressions
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:
![](./articles_imgs/5958/radica15.gif)
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
![](./articles_imgs/5958/radica19.jpg)
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 ![](./articles_imgs/5958/radica21.gif)
is asking ![](./articles_imgs/5958/radica23.gif)
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
![](./articles_imgs/5958/radica25.gif)
is asking ![](./articles_imgs/5958/radica27.gif)
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 ![](./articles_imgs/5958/radica29.gif)
2. Evaluate ![](./articles_imgs/5958/radica30.gif)
3. Evaluate ![](./articles_imgs/5958/radica31.gif)
4. Evaluate ![](./articles_imgs/5958/radica32.gif)
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 ![](./articles_imgs/5958/radica35.gif)
If is a real number, then
![](./articles_imgs/5958/radica37.gif)
where
is the exponential
form of the expression, and
is the radical form of the
expression.
Example 2
Put each expression in radical form.
![](./articles_imgs/5958/radica40.jpg)
Example 3
Put each expression in exponential form.
![](./articles_imgs/5958/radica41.jpg)
Example 4
Simplify each expression (reduce the index).
![](./articles_imgs/5958/radica42.jpg)
Product Property of Radicals
If and
are real numbers, then
![](./articles_imgs/5958/radica43.gif)
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
![](./articles_imgs/5958/radica46.jpg) |
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 ![](./articles_imgs/5958/radica48.gif)
Example 2
Since n = 3, the radicand a can have no perfect cube
factors.
Simplify
Simplify ![](./articles_imgs/5958/radica50.gif)
Example 3
Since n = 4, the radicand a can have no 4th-power factors.
Simplify
Simplify ![](./articles_imgs/5958/radica52.gif)
Example 4 Simplify each expression.
![](./articles_imgs/5958/radica53.gif)
![](./articles_imgs/5958/radica54.gif)
Example 5 Simplify each expression.
![](./articles_imgs/5958/radica55.gif)
![](./articles_imgs/5958/radica56.gif)
![](./articles_imgs/5958/radica57.gif)
![](./articles_imgs/5958/radica58.gif)
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.
![](./articles_imgs/5958/radica60.gif)
![](./articles_imgs/5958/radica61.gif)
![](./articles_imgs/5958/radica62.gif)
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