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CHAPTER R:
Basic Concepts of Algebra
R.1 The Real-Number System
R.2 Integer Exponents, Scientific Notation, and Order of
Operations
R.3 Addition, Subtraction, and Multiplication of Polynomials
R.4 Factoring
R.5 The Basics of Equation Solving
R.6 Rational Expressions
R.7 Radical Notation and Rational Exponents
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
R.7
Radical Notation and Rational
Exponents

Simplify radical expressions.
 Rationalize denominators or numerators in
rational expressions.
 Convert between exponential and radical
notation.
 Simplify expressions with rational exponents.
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Notation
A number c is said to be a square root of a if c2 = a.
Similarly, c is a cube root of a if c3 = a.
nth Root
A number c is said to be an nth root of a if cn = a.
The symbol n a denotes the nth root of a. The symbol is
called a radical. The number n is called the index.
Any positive number has two square roots, one positive and
one negative. For any even index, a positive number has two
real-number roots. The positive root is called the principal
root.
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Slide R.7 - 4
Examples
Simplify each of the following:
a) 49 = 7, because 72 = 49.
b)  49 = 7, because 72 = 49 and 


49  (7)  7.
3
3
3
3
27
27
3




.
c) 3
because

 
3
125 5
 5  5 125
d)
e)
3
4
64  4 because (4)3 = 64.
25 is not a real number.
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Slide R.7 - 5
Properties of Radicals
Let a and b be any real numbers or expressions for
which the given roots exist. For any natural numbers
m and n (n  1):
1. If n is even, n a n  a
2. If n is odd, n a n  a
3. n a  n b  n ab
4. n a  n a (b  0)
5.
n
b
n
a 
 a
m
b
n
m
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Slide R.7 - 6
Examples
a) (7)  7  7
2
b)
c)
d)
e)
3
4
(7)3  7
3
f)
27 
5

3
27

5
 35  243
245 x 6 y 5  49  5  x 6  y 4  y
g)
 49 x 6 y 4 5 y
3  4 7  4 21
 7 x3 y 2
5 y  7 y 2 x3
5y
72  36  2  6 2
81
9

81
 9 3
9
h)
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
y
y2
y2


25
25 5
Slide R.7 - 7
Another Example
Perform the operation.
3
2 3

  2
2 3 3 3
2
9 6  6 3
 3
2
 3  2  (9  1) 6  3  3
 68 6 9
 3  8 6
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Slide R.7 - 8
The Pythagorean Theorem
The sum of the squares of
the lengths of the legs of a
right triangle is equal to
the square of the length of
the hypotenuse:
a2 + b2 = c2.
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
c
a
b
Slide R.7 - 9
Example
Juanita paddled her canoe across a river 525 feet wide.
A strong current carried her canoe 810 feet
downstream as she paddled. Find the distance Juanita
actually paddled, to the nearest foot.
Solution:
810 ft
c2  a 2  b2
c  5252  8102
525 ft
x
c  275,625  656,100
c  931,725
c  965.3 ft
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Slide R.7 - 10
Rationalizing Denominators or Numerators
Rationalizing the denominator (or numerator) is
done by multiplying by 1 in such a way as to obtain
a perfect nth power.
Example
Rationalize the denominator.
6
6 7
42
42

 

7
7 7
7
49
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Slide R.7 - 11
Rationalizing Denominators or Numerators
Conjugates
Pairs of expressions of the form
a b  c d and a b  c d
are called conjugates. The product of a pair
contains no radicals and can be used to rationalize a
denominator or numerator.
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Slide R.7 - 12
Rationalizing Denominators or Numerators
Example
Rationalize the numerator.
a b
a b a b


4
4
a b
a  b


2

2
4 a 4 b
a b
4 a 4 b
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Slide R.7 - 13
Rational Exponents
For any real number a and any natural numbers m and
n, n  1, for which n a exists,
1/ n
a
a
a
m/n
 n a,
 a 
m / n
m
n

1
a
m/n
 a
n
m
, and
.
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Slide R.7 - 14
Examples
Convert to radical notation and, if possible, simplify.
a) 113/ 4  4 113
b) 91/ 2 
1
1 1


1/ 2
9
9 3
c)  27 4 / 3  3 (27)4
 27 
4/3


3
27

4
 3 531, 441  81, or
 (3) 4  81
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Slide R.7 - 15
More Examples
Convert each to exponential notation.
6
5
a)  8ab   8ab 6 / 5
b)
12
x 4  x 4 /12  x1/ 3
Simplify.
a) x7 / 8  x3/ 4  x7 / 83/ 4  x13/ 8  8 x13  8 x8  8 x5  x
b)
8
x5
( x  2)7 / 3 ( x  2)1/ 3  ( x  2)7 / 31/ 3  ( x  2)2
Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley
Slide R.7 - 16