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Section P.3

Prerequisites

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

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

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

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

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

P.3

For any positive integer n and real numbers a and b (b ¹ 0),

provided that all of the roots are real.

nnn baab

n

n

n

ba

ba

The Product and QuotientRules for Radicals

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

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

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

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