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Copyright © 2011 Pearson Education, Inc.
Rational Exponents and Radicals
Section P.3
Prerequisites
Copyright © 2011 Pearson Education, Inc. Slide P-3
P.3
Definition: nth RootsIf n is a positive even integer and an = b, then a is called an nth root of b.
If a2 = b, then a is a square root of b.If a3 = b, then a is a cube root of b.
If n is even (or odd) and a is an nth root of b, then a is called an even (or odd) root of b.
Every positive real number has two real even roots, a positive root and a negative root.
Every real number has exactly one real odd root.
Roots
Copyright © 2011 Pearson Education, Inc. Slide P-4
P.3
Definition: Exponent 1/nIf n is a positive even integer and a is positive, then a1/n
denotes the positive real nth root of a and is called the principal nth root of a.
If n is a positive odd integer and a is any real number, then a1/n denotes the real nth root of a.
If n is a positive integer, then 01/n = 0.
Definition: Rational ExponentsIf m and n are positive integers, then am/n = (a1/n)m, provided that a1/ n is a real number.
Note that a1/ n is not real when a is negative and n is even.
Roots and Rational Exponents
Copyright © 2011 Pearson Education, Inc. Slide P-5
P.3
Procedure: Evaluating a–m/n
To evaluate a–m/n mentally, 1. find the nth root of a,2. raise it to the m power, and3. find the reciprocal.
Rational Exponents
Copyright © 2011 Pearson Education, Inc. Slide P-6
P.3
The following rules are valid for all real numbers a and b and rational numbers r and s, provided that all indicated powers are real and no denominator is zero.
1. 2. 3.
4. 5. 6.
7.
srsr aaa
rrr baab )(
sr
s
r
aaa
r
rr
ba
ba
r
s
s
r
ab
ba
srsr aa )(
rr
ab
ba
Rules for Rational Exponents
Copyright © 2011 Pearson Education, Inc. Slide P-7
P.3
Definition: RadicalIf n is a positive integer and a is a number for which a1/n is defined, then the expression is called a radical, and
If n = 2, we write rather than
Rule: Converting am/n to Radical NotationIf a is a real number and m and n are integers for which is real, then
n a
a
n a
./1 nn aa
.2 a
./ n mmnnm aaa
Radical Notation
Copyright © 2011 Pearson Education, Inc. Slide P-8
P.3
For any positive integer n and real numbers a and b (b ¹ 0),
1. Product rule for radicals
2. Quotient rule for radicals
provided that all of the roots are real.
nnn baab
n
n
n
ba
ba
The Product and QuotientRules for Radicals
Copyright © 2011 Pearson Education, Inc. Slide P-9
P.3
An expression that is the square of a term that contains no radicals is called a perfect square.
An expression that is the cube of a term that contains no radicals is called a perfect cube.
In general, an expression that is the nth power of an expression that contains no radicals is a perfect nth power.
The product and quotient rules for radicals are used to simplify radicals containing perfect squares, perfect cubes, and so on.
The Product and QuotientRules for Radicals
Copyright © 2011 Pearson Education, Inc. Slide P-10
P.3
The product rule is used to remove the perfect nth powers that are factors of the radicand, and the quotient rule is used when fractions occur in the radicand.
The process of removing radicals from the denominator is called rationalizing the denominator.
Since radical expressions with the same index are added in the same manner as variable like terms, they are called like terms or like radicals.
Rationalizing the Denominator
Copyright © 2011 Pearson Education, Inc. Slide P-11
P.3
Definition: Simplified Form for Radicals of Index nA radical of index n in simplified form has
1. no perfect nth powers as factors of the radicand,
2. no fractions inside the radical, and
3. no radicals in a denominator.
Theorem: mth Root of an nth RootIf m and n are positive integers for which all of the following roots are real, then
.nmm n aa
Simplified Form andRationalizing the Denominator