Radical Expressions and Rational Exponents

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Radical Expressions and Rational Exponents
Warm Up
Lesson Presentation
Lesson Quiz
Holt ALgebra2
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Warm Up Simplify each expression.
16,807
1. 73 72
121
729
3. (32)3
4.
5.
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Objectives
Rewrite radical expressions by using rational
exponents. Simplify and evaluate radical
expressions and expressions containing rational
exponents.
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Vocabulary
index rational exponent
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You are probably familiar with finding the square
root of a number. These two operations are
inverses of each other. Similarly, there are
roots that correspond to larger powers.
5 and 5 are square roots of 25 because 52 25
and (5)2 25
2 is the cube root of 8 because 23 8.
2 and 2 are fourth roots of 16 because 24 16
and (2)4 16.
a is the nth root of b if an b.
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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.
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Example 1 Finding Real Roots
Find all real roots.
A. sixth roots of 64
A positive number has two real sixth roots.
Because 26 64 and (2)6 64, the roots are 2
and 2.
B. cube roots of 216
A negative number has one real cube root. Because
(6)3 216, the root is 6.
C. fourth roots of 1024
A negative number has no real fourth roots.
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Check It Out! Example 1
Find all real roots.
a. fourth roots of 256
A negative number has no real fourth roots.
b. sixth roots of 1
A positive number has two real sixth roots.
Because 16 1 and (1)6 1, the roots are 1 and
1.
c. cube roots of 125
A positive number has one real cube root. Because
(5)3 125, the root is 5.
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The properties of square roots in Lesson 1-3 also
apply to nth roots.
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Example 2A Simplifying Radical Expressions
Simplify each expression. Assume that all
variables are positive.
Factor into perfect fourths.
Product Property.
3 ? x ? x ? x
Simplify.
3x3
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Example 2B Simplifying Radical Expressions
Quotient Property.
Simplify the numerator.
Rationalize the numerator.
Product Property.
Simplify.
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Check It Out! Example 2a
Simplify the expression. Assume that all
variables are positive.
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4
16
x
Factor into perfect fourths.
Product Property.
2 ? x
Simplify.
2x
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Check It Out! Example 2b
Simplify the expression. Assume that all
variables are positive.
Quotient Property.
Rationalize the numerator.
Product Property.
Simplify.
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Check It Out! Example 2c
Simplify the expression. Assume that all
variables are positive.

Product Property of Roots.
x3
Simplify.
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Example 3 Writing Expressions in Radical Form
Method 1 Evaluate the root first.
Method 2 Evaluate the power first.
Write with a radical.
Write with a radical.
(2)3
Evaluate the root.
Evaluate the power.
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Evaluate the power.
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Evaluate the root.
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Check It Out! Example 3a
Write the expression in radical form, and
simplify.
Method 1 Evaluate the root first.
Method 2 Evaluate the power first.
Write with a radical.
Write will a radical.
(4)1
Evaluate the root.
Evaluate the power.
4
Evaluate the power.
4
Evaluate the root.
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Check It Out! Example 3b
Write the expression in radical form, and
simplify.
Method 1 Evaluatethe root first.
Method 2 Evaluatethe power first.
Write with a radical.
Write with a radical.
(2)5
Evaluate the root.
Evaluate the power.
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Evaluate the power.
32
Evaluate the root.
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Check It Out! Example 3c
Write the expression in radical form,
and simplify.
Method 1 Evaluatethe root first.
Method 2 Evaluate the power first.
Write with a radical.
Write with a radical.
(5)3
Evaluate the root.
Evaluate the power.
125
Evaluate the power.
125
Evaluate the root.
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Example 4 Writing Expressions by Using Rational
Exponents
Write each expression by using rational exponents.
Simplify.
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Simplify.
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Check It Out! Example 4
Write each expression by using rational exponents.
a.
b.
c.
103
Simplify.
Simplify.
1000
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Rational exponents have the same properties as
integer exponents (See Lesson 1-5)
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Example 5A Simplifying Expressions with Rational
Exponents
Simplify each expression.
Product of Powers.
Simplify.
72
Evaluate the Power.
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Check Enter the expression in a graphing
calculator.
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Example 5B Simplifying Expressions with Rational
Exponents
Simplify each expression.
Quotient of Powers.
Simplify.
Negative Exponent Property.
Evaluate the power.
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Example 5B Continued
Check Enter the expression in a graphing
calculator.
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Check It Out! Example 5a
Simplify each expression.
Product of Powers.
Simplify.
Evaluate the Power.
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Check Enter the expression in a graphing
calculator.
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Check It Out! Example 5b
Simplify each expression.
Negative Exponent Property.
Evaluate the Power.
Check Enter the expression in a graphing
calculator.
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Check It Out! Example 5c
Simplify each expression.
Quotient of Powers.
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Simplify.
Evaluate the power.
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Check Enter the expression in a graphing
calculator.
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Example 6 Chemistry Application
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Example 6 Continued
Substitute 800 for t.
Simplify.
Negative Exponent Property.
Simplify.
Use a calculator.
354
The amount of radium-226 left after 800 years
would be about 354 mg.
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Check It Out! Music Application
Use 64 cm for the length of the string, and
substitute 12 for n.
64(21)
Simplify.
Negative Exponent Property.
Simplify.
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The fret should be placed 32 cm from the bridge.
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Lesson Quiz Part I
Find all real roots.
5, 5
1. fourth roots of 625
2. fifth roots of 243
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Simplify each expression.
4.
4y2
8
3.
256
y
4
2
6. Write using rational exponents.
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Lesson Quiz Part II
7. If 2000 is invested at 4 interest compounded
monthly, the value of the investment after t
years is given by . What is the
value of the investment after 3.5 years?
2300.01