Chapter R Section 8: nth Roots and
Rational Exponents
In this section, we will…
Evaluate nth Roots
Simplify Radical Expressions
Add, Subtract, Multiply and
Divide Radical Expressions
Rationalize Denominators
Simplify Expressions with Rational Exponents
Factor Expressions with Radicals or Rational Exponents
R.8 nth Roots and Rational Expressions
Recall from
Review
Section 2…
2
If a is a non-negative real number, any number b, such that b  a is the
square root of a and is denoted b  a
If a is a non-negative real number, any non-negative number b, such that
b 2  a is the primary square root of a and is denoted b  a
Examples: Evaluate the following by taking the square root.
121
The principal root of a positive number is positive
principal root:
16
Negative numbers do not have real # square roots
R.8 nth Roots and Rational Expressions: nth Roots
The principal nth root of a real number a, n > 2 an integer, symbolized by
a is defined as follows: n a  b means a  bn
 where a  0 and b  0if n is even
 where a, b are any real number if n is odd
n
index
n
a
radicand
radical
Examples: Simplify each expression.
3
27
3
8
4
81
principal root:
4
16
5
1
R.8 nth Roots and Rational Expressions: nth Roots
n
a n  a if n  2 is even
n
an  a if n  3 is odd
Properties of Radicals: Let n  2 and m  2 denote positive integers and let
a and b represent real numbers. Assuming that all radicals are defined:
n
ab  a b
n
n
n
a na
 n
b
b
n
a 
m
 a
n
Simplifying Radicals: A radical is in simplest form when:
 No radicals appear in the denominator of a fraction
 The radicand cannot have any factors that are perfect roots
(given the index)
Examples: Simplify each expression.
12
50
3
16
R.8 nth Roots and Rational Expressions: Simplify Radical Expressions
m
Simplifying Radical Expressions Containing Variables:
Examples: Simplify each expression. Assume that all variables are positive.
5
x5
4
16x8
b7
3
When we divide the exponent by
the index, the remainder remains
under the radical
54x 6 y 5
R.8 nth Roots and Rational Expressions: Simplify Radical Expressions
Adding and Subtracting Radical Expressions:
 simplify each radical expression
 combine all like-radicals
(combine the coefficients and keep the common radical)
Examples: Simplify each expression. Assume that all variables are positive.
125  20
2 12  3 27
8 xy  25 x 2 y 2  3 8 x 3 y 3
4
32 x  4 2 x5
R.8 nth Roots and Rational Expressions: Add, Subtract, Multiply and Divide Radicals
Multiplying and Dividing Radical Expressions:
Examples: Simplify each expression. Assume that all variables are positive.
5x 20 x
3
3

3
3 xy 2
we will use: n ab  n a n b
we will use:
n a
b

n
n
81x 4 y 2
3
3 10

4
we will use: a 
n
m
a
b
 a
n
m
R.8 nth Roots and Rational Expressions: Add, Subtract, Multiply and Divide Radicals
Examples: Simplify each expression. Assume that all variables are positive.
5 8  3 3 
4 2 3 5 2 8




2
x 2

3 5

x 2
2
R.8 nth Roots and Rational Expressions: Add, Subtract, Multiply and Divide Radicals
Rationalizing Denominators: Recall that simplifying a radical expression
means that no radicals appear in the denominator of a fraction.
Examples: Simplify each expression. Assume that all variables are positive.
24
5
5
4 2
4
3
2
R.8 nth Roots and Rational Expressions: Rationalize Denominators
Rationalizing Binomial Denominators:
example:
2
3 1
The conjugate of the binomial a + b is a – b and the conjugate of a – b is a + b.
Examples: Simplify each expression. Assume that all variables are positive.
2
3 1
2 1
3 22
R.8 nth Roots and Rational Expressions: Rationalize Denominators
Evaluating Rational Exponents:
If a is a real number and n  2 is an integer and assuming that all radicals are defined:
1
n
a na
Examples: Simplify each expression.
16
1
4
0
1
8
If a is a real number and m and n  2 are an integer and assuming that all radicals
m
m
are defined:
n
m
n
n
a  a 
4
 
3
2
25
 a
27
8
2
3
 32
R.8 nth Roots and Rational Expressions: Simplify Expressions with Rational Exponents
Simplifying Expressions Containing Rational Exponents: Recall the
following from Review Section 2:
Laws of Exponents: For any integers m, n (assuming no divisions by 0)
xm
mn
m n
mn

x
x x x
xn
n
n
n
n n


x
x
 xy   x y
   n
y
 y
x
x
n

m n
1
 n
x
and
1
n
x0  1
 x mn
a  a
n
x
 
 y
1
n

x
xn
m
n
n
 y
 
x
a  a 
n
m
n
 a
n
m
R.8 nth Roots and Rational Expressions: Simplify Expressions with Rational Exponents
Examples: Simplify each expression. Express your answer so that only
positive exponents occur. Assume that the variables are positive.
2
3
1
2
x x x
 14
x y 
4
8
3
4
x y 
 x y
 xy 
1
4
2
2
2
1
2
3
4
R.8 nth Roots and Rational Expressions: Simplify Expressions with Rational Exponents
Factoring Expressions with Radicals and/or Rational Exponents:
Recall that, when factoring, we take out the GCF with the smallest
exponent in the terms.
Examples: Factor each expression. Express your answer so that only positive
exponents occur.
10 x2  x  1  5x  x  1
3
2
x x
1
2
 x  3   x  3 
3
2
1
2
R.8 nth Roots and Rational Expressions: Factor Expressions with Radicals/Rational Exponents
Examples: Factor each expression. Express your answer so that only positive
exponents occur.
1
4 2
3
x

4

x

x

4
 2x




3
2
4
3
R.8 nth Roots and Rational Expressions: Factor Expressions with Radicals/Rational Exponents
Examples: Factor each expression. Express your answer so that only positive
exponents occur.
6 x  2 x  3  x  8
1
2
3
2
x0
R.8 nth Roots and Rational Expressions: Factor Expressions with Radicals/Rational Exponents
Examples: Factor each expression. Express your answer so that only positive
exponents occur.
2 x  3x  4   x  4  3x  4 
4
3
2
1
3
R.8 nth Roots and Rational Expressions: Factor Expressions with Radicals/Rational Exponents
Example: The final velocity, v, of an object in feet per second (ft/sec)
after it slides down a frictionless inclined plane of height h feet is:
where v0 is the initial velocity
2
v  64h  v0 in ft/sec of the object.
What is the final velocity, v, of an object that slides down a frictionless
inclined plane of height 2 feet with an initial velocity of 4 ft/sec?
R.8 nth Roots and Rational Expressions: Applications
Independent Practice
You learn math by doing math. The best way to learn math is to practice,
practice, practice. The assigned homework examples provide you with an
opportunity to practice. Be sure to complete every assigned problem (or more
if you need additional practice). Check your answers to the odd-numbered
problems in the back of the text to see whether you have correctly solved each
problem; rework all problems that are incorrect.
Read pp. 72-76
Homework:
pp. 77-79 #7-51 odds, 55-73 odds,
89-93 odds, 107
R.8 nth Roots and Rational Expressions
Review of Exam Policies and Procedures
Page 7 of the Student Guide and Syllabus
From Math for Artists…
“These are the laws of exponents and radicals in bright, cheerful, easy to memorize colors.”
R.8 nth Roots and Rational Expressions