Algebra II Chapter 6 Notes and Practice – Packet 1-Evaluation 1
Radical Functions and Rational ExponentsNth Roots of an expression
The square root of an expression is that expression you can square that would produce the
expression under the square root.
16  4
4 is the square root of 16 because 4 2  16
Similar to square roots, a cube root of an expression is an expression you can cube and end up
with the expression under the cube root.
3
27  3
3 is the cube root of 27 because 33  27 .
This concept can be generalized to apply to any positive integer root of an expression.
Nth Roots
For any real numbers “a” and “b”, and any positive integer “n”, If a n  b then “a” is
considered the nth root of “b”.
For any even roots of positive numbers there are two roots one positive and one negative..
For example the 4th root of 16 is either 2 or –2 because both of those numbers to
the 4th power equal positive 16.
There are no even roots of negative numbers because any real number raised to an even power
must be positive.
For any odd roots of a positive number there is only one positive root.
For example the cube root of 8 is only 2 because only 2 cubed gives you 8.
For any odd roots of a negative number there is only one negative root also.
For example the fifth root of –32 is –2 because (-2) to the fifth power is –32.
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Radical Notation
Radical notation in math is any positive integer root of a real number written in
the form
n
a b
n is the root (index) of “a”(radicand) such that a n  b (principle
root)
Example
4
16  2
2 is said to be the principle 4th root of 16.
- 4 16  2
Nth roots of a n , a  0
For any negative real number a,
n
a n  a when n is even
Simplifying Radical expressions
Examples
Simplify the following radical expressions
Example 1
4
16 x8  4 24 ( x 4 ) 2  2 x 2
Note: since x 2 can never be negative absolute value symbols are not needed around the
x 2 in the answer.
9 x 6  32 ( x3 )2  3 x3 Note: since the square root of x 6  x3 and it is possible for x 3 to
be a negative number, absolute value symbols are necessary.
NOTE: ANY TIME YOU TAKE THE EVEN ROOT OF A VARIABLE RAISED TO AN
EVEN POWER AND THE RESULT LEAVES YOU WITH AN ODD EXPONENT,
ABSOLUTE VALUE SYMBOLS ARE NEEDED.
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Practice Problems
Simplify each of the given Radical Expressions. If absolute value symbols are needed
make sure you use them. Find only real roots.
1.
81
5.
16x 2 y 2
2.
 25
3.
4
16
x 8 y18
6.
4.
7.
3
3
27
27 y 6
Multiplying and Dividing Radical Expressions
Radical expressions can be multiplied or divided as long as the index of each
root is the same.
Multiplication of nth roots
If n a and
n
b are real numbers then
n
a  n b  n ab
Examples
Simplify the following Products if possible
Ex: 1
18  2
36  6
3
3x5  3 9 y 4
3
27 x5 y 4  3xy 3 x 2 y
Ex: 2
-3-
Ex: 3
8  2
16  4
4  2
Ex: 4 8
Not a real number
Practice Problems
Simplify the given products if possible
1. 3  12
2. 3 4x3  3 16x7
Dividing Radical Expressions
3.
5  5
4.
3 x3  3 y 5
If two radical expressions with the same index are being divided, you can
rewrite the division as a single radical expression with the quotient on the inside of the
radical with the given index.
n
n
a na

for b  0.
b
b
Examples of Division of radicals (Assume all variables are positive)
8
8

 42
2
2
Ex: 1
Ex: 3
4
28 x7
4
14 x 2
4
Ex: 2
16 x3
16 x3

 4x2  2x
4
x
4x
28 x7 4 5
 2x  x 4 2x
2
14 x
Rationalizing the Denominator
Sometimes when you divide radical expressions you are left with a radical
expression in the denominator of the fraction or you are left with a denominator inside
of a radical. Neither case is desirable in an expression. So to eliminate this problem
you will need to do a process called rationalizing a denominator. Examples of
rationalizing the denominator are shown below.
Examples: Rationalize the denominator of each expression. Assume that all
variables are positive.
Ex:
To rationalize a square root in a denominator multiply the top and bottom
of the fraction by the denominator After you simplify the fraction.
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5
3
Method 1.
Ex:
3 5 3

3
3
5 x3
5
5


4
2x
2x
2x
Method 1. Ex 2
2x
10 x

2x
2x
Rationalizing a denominator that is an “nth root” requires you to multiply
the numerator and denominator by something that makes the denominator
a perfect “nth root”.
Method 2 (Rationalizing nth roots)
3
x4
x4
x3
3
3
3 5 xy
5 xy
5y
Ex: 3
4
Ex: 4
4
6 x3 y
6 x3 y


2 x5 y 2
2 x5 y 2
4
4
52 y 2 3 25 x3 y 2 x 3 25 y 2


3 3 3
52 y 2
5y
5 y
3
x2 y
4
x2 y3
4
x2 y3

4
4
3x 2 y 3
x4 y 4
3x 2 y 3

xy
4
Practice Problems - Divide the following radical expressions. If necessary
rationalize the denominator.
81
3
3
3
1.
3
4.
3
5 x3
15 x 2
2.
5.
56 x 5 y 5
7 xy
5
3 2x
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3.
4
16 x 7 y 5
4 x3 y
5
16 x5 y 2
5
4 x6 y8
6.
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