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Holt McDougal Algebra 2
5-6 Radical Expressions and Rational Exponents 5-6 Radical Expressions and Rational Exponents
Holt ALgebra2
Warm UpLesson PresentationLesson Quiz
Holt McDougal Algebra 2
Holt McDougal Algebra 2
5-6 Radical Expressions and Rational Exponents
Warm Up
Simplify each expression.16,807
121
729
1. 73 • 72
3. (32)3
2. 118
116
4.
5.
75207
2 357
5 3
Holt McDougal Algebra 2
5-6 Radical Expressions and Rational Exponents
Rewrite radical expressions by using rational exponents.Simplify and evaluate radical expressions and expressions containing rational exponents.
Objectives
Holt McDougal Algebra 2
5-6 Radical Expressions and Rational Exponents
indexrational exponent
Vocabulary
Holt McDougal Algebra 2
5-6 Radical Expressions and Rational Exponents
The nth root of a real number a can be written as the radical expression , where n is the index (plural: indices) of the radical and a is the radicand. When a number has more than one root, the radical sign indicates only the principal, or positive, root.
n a
Holt McDougal Algebra 2
5-6 Radical Expressions and Rational Exponents
When a radical sign shows no index, it represents a square root.
Reading Math
Holt McDougal Algebra 2
5-6 Radical Expressions and Rational Exponents
Find all real roots.Example 1: Finding Real Roots
A. sixth roots of 64A positive number has two real sixth roots. Because 26 = 64 and (–2)6 = 64, the roots are 2 and –2.
B. cube roots of –216A negative number has one real cube root. Because (–6)3 = –216, the root is –6.
C. fourth roots of –1024A negative number has no real fourth roots.
Holt McDougal Algebra 2
5-6 Radical Expressions and Rational Exponents
Find all real roots.a. fourth roots of –256
A negative number has no real fourth roots.
Check It Out! Example 1
b. sixth roots of 1A positive number has two real sixth roots. Because 16 = 1 and (–1)6 = 1, the roots are 1 and –1.
c. cube roots of 125A positive number has one real cube root. Because (5)3 = 125, the root is 5.
Holt McDougal Algebra 2
5-6 Radical Expressions and Rational Exponents
The properties of square roots in Lesson 1-3 also apply to nth roots.
Holt McDougal Algebra 2
5-6 Radical Expressions and Rational Exponents
When an expression contains a radical in the denominator, you must rationalize the denominator. To do so, rewrite the expression so that the denominator contains no radicals.
Remember!
Holt McDougal Algebra 2
5-6 Radical Expressions and Rational Exponents
Simplify each expression. Assume that all variables are positive.
Example 2A: Simplifying Radical Expressions
Factor into perfect fourths.
Product Property.
Simplify.
3 x x x 3x3
Holt McDougal Algebra 2
5-6 Radical Expressions and Rational Exponents
Simplify the expression. Assume that all variables are positive.
Check It Out! Example 2a
Product Property.
Simplify.
Factor into perfect fourths.
2 x2x
4 416x
4 24 •4 x4
4 24 •x4
Holt McDougal Algebra 2
5-6 Radical Expressions and Rational Exponents
Check It Out! Example 2c
3 37 2x x
Product Property of Roots.
Simplify.
Simplify the expression. Assume that all variables are positive.
x3
3 9x
Holt McDougal Algebra 2
5-6 Radical Expressions and Rational Exponents
A rational exponent is an exponent that can be expressed as , where m and n are integers and n ≠ 0. Radical expressions can be written by using rational exponents.
mn
Holt McDougal Algebra 2
5-6 Radical Expressions and Rational Exponents
The denominator of a rational exponent becomes the index of the radical.
Writing Math
Holt McDougal Algebra 2
5-6 Radical Expressions and Rational Exponents
Example 3: Writing Expressions in Radical Form
Method 1 Evaluate the root first.
(–2)3
Write with a radical.
Write the expression (–32) in radical form and simplify.
35
–8
Evaluate the root.
Evaluate the power.
Method 2 Evaluate the power first.
Write with a radical.
–8
Evaluate the power.
Evaluate the root.
( )-3
5 32
-5 32,768
Holt McDougal Algebra 2
5-6 Radical Expressions and Rational Exponents
Method 1 Evaluate the root first.
(4)1
Write with a radical.
6413
4
Evaluate the root.
Evaluate the power.
Check It Out! Example 3a Write the expression in radical form, and simplify.
Method 2 Evaluate the power first.
Write will a radical.
4
Evaluate the power.
Evaluate the root.
( )13 64 ( )13 64
3 64
Holt McDougal Algebra 2
5-6 Radical Expressions and Rational Exponents
Method 1 Evaluatethe root first.
(2)5
Write with a radical.
452
32
Evaluate the root.
Evaluate the power.
Check It Out! Example 3b
Write the expression in radical form, and simplify.
Method 2 Evaluatethe power first.
Write with a radical.
32
Evaluate the power.
Evaluate the root.
( )52 4 ( )52 4
2 1024
Holt McDougal Algebra 2
5-6 Radical Expressions and Rational Exponents
Method 1 Evaluatethe root first.
(5)3
Write with a radical.
62534
125
Evaluate the root.
Evaluate the power.
Check It Out! Example 3c Write the expression in radical form, and simplify.
Method 2 Evaluate the power first.
Write with a radical.
125
Evaluate the power.
Evaluate the root.
( )34 625 ( )34 625
4 244,140,625
Holt McDougal Algebra 2
5-6 Radical Expressions and Rational Exponents
Example 4: Writing Expressions by Using Rational Exponents
Write each expression by using rational exponents.
Simplify.
15 5 3
13 12 Simplify.
A. B.
4813
33
27
=m
mn na a =m
mn na a
Holt McDougal Algebra 2
5-6 Radical Expressions and Rational Exponents
Write each expression by using rational exponents.
Simplify.
a. b.
3481
103
Check It Out! Example 4
9310
1000
=m
mn na a=m
mn na a
Simplify. 5 12
c.
24 5 =
mmn na a
Holt McDougal Algebra 2
5-6 Radical Expressions and Rational Exponents Rational exponents have the same properties as integer exponents (See Lesson 1-5)
Holt McDougal Algebra 2
5-6 Radical Expressions and Rational Exponents
Example 5A: Simplifying Expressions with Rational Exponents
Product of Powers.
Simplify each expression.
Simplify.
Evaluate the Power.
72 49
Check Enter the expression in a graphing calculator.
Holt McDougal Algebra 2
5-6 Radical Expressions and Rational Exponents
Example 5B: Simplifying Expressions with Rational Exponents
Quotient of Powers.
Simplify each expression.
Simplify.
Negative Exponent Property.
1 4
Evaluate the power.