3-4: Rational Exponents and Radical Equations English Casbarro Unit 3.
Holt Algebra 2 8-6 Radical Expressions and Rational Exponents 8-6 Radical Expressions and Rational...
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Holt Algebra 2
8-6 Radical Expressions and Rational Exponents 8-6 Radical Expressions and Rational Exponents
Holt ALgebra2
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2
8-6 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.
75
20
7
2 357
5 3
Holt Algebra 2
8-6 Radical Expressions and Rational Exponents
Rewrite radical expressions by using rational exponents.
Simplify and evaluate radical expressions and expressions containing rational exponents.
Objectives
Holt Algebra 2
8-6 Radical Expressions and Rational Exponents
indexrational exponent
Vocabulary
Holt Algebra 2
8-6 Radical Expressions and Rational Exponents
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.
Holt Algebra 2
8-6 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 Algebra 2
8-6 Radical Expressions and Rational Exponents
When a radical sign shows no index, it represents a square root.
Reading Math
Holt Algebra 2
8-6 Radical Expressions and Rational Exponents
Find all real roots.
Example 1: Finding 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.
Holt Algebra 2
8-6 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 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.
Holt Algebra 2
8-6 Radical Expressions and Rational Exponents
The properties of square roots in Lesson 1-3 also apply to nth roots.
Holt Algebra 2
8-6 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 Algebra 2
8-6 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 Algebra 2
8-6 Radical Expressions and Rational Exponents
Example 2B: Simplifying Radical Expressions
Quotient Property.
Product Property.
Simplify the numerator.
Rationalize the numerator.
Simplify.
Holt Algebra 2
8-6 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 x
2x
4 416x
4 24 •4 x4
4 24 •x4
Holt Algebra 2
8-6 Radical Expressions and Rational Exponents
Check It Out! Example 2b
Quotient Property.
Product Property.
Rationalize the numerator.
Simplify.
Simplify the expression. Assume that all variables are positive.
84
4 3
x
4227
3x
Holt Algebra 2
8-6 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 Algebra 2
8-6 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 Algebra 2
8-6 Radical Expressions and Rational Exponents
The denominator of a rational exponent becomes the index of the radical.
Writing Math
Holt Algebra 2
8-6 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 Algebra 2
8-6 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 Algebra 2
8-6 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 Algebra 2
8-6 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 Algebra 2
8-6 Radical Expressions and Rational Exponents
Example 4: Writing Expressions by Using Rational Exponents
Write each expression by using rational exponents.
Simplify.
15 5 3
1312 Simplify.
A. B.
4813
33
27
=m
mn na a =m
mn na a
Holt Algebra 2
8-6 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 am
mn na a
Simplify. 512
c.
24 5
mmn na a
Holt Algebra 2
8-6 Radical Expressions and Rational Exponents
Rational exponents have the same properties as integer exponents (See Lesson 1-5)
Holt Algebra 2
8-6 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 Algebra 2
8-6 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.
Holt Algebra 2
8-6 Radical Expressions and Rational Exponents
Example 5B Continued
Check Enter the expression in a graphing calculator.
Holt Algebra 2
8-6 Radical Expressions and Rational Exponents
Product of Powers.
Simplify each expression.
Simplify.
Evaluate the Power.6
Check It Out! Example 5a
Check Enter the expression in a graphing calculator.
Holt Algebra 2
8-6 Radical Expressions and Rational Exponents
Simplify each expression.
(–8)–13
Check It Out! Example 5b
1 –8
13
1 2
–
Negative Exponent Property.
Evaluate the Power.
Check Enter the expression in a graphing calculator.
Holt Algebra 2
8-6 Radical Expressions and Rational Exponents
Quotient of Powers.
Simplify each expression.
Simplify.
Evaluate the power.
52
Check It Out! Example 5c
25
Check Enter the expression in a graphing calculator.
Holt Algebra 2
8-6 Radical Expressions and Rational Exponents
Example 6: Chemistry Application
Radium-226 is a form of radioactive element
that decays over time. An initial sample of
radium-226 has a mass of 500 mg. The mass of
radium-226 remaining from the initial sample
after t years is given by . To the
nearest milligram, how much radium-226 would
be left after 800 years?
Holt Algebra 2
8-6 Radical Expressions and Rational Exponents
Example 6 Continued
Substitute 800 for t.
Simplify.
Negative Exponent Property.
800 1600500 (200– ) = 500(2– )
t1600
= 500( ) 1
2 12
12 = 500(2– )
= 500
2 12
Simplify.
Use a calculator.≈ 354The amount of radium-226 left after 800 years would be about 354 mg.
Holt Algebra 2
8-6 Radical Expressions and Rational Exponents
Negative Exponent Property.
Use 64 cm for the length of the string, and substitute 12 for n.
Check It Out! Music Application
= 64(2–1)
Simplify.= 32
The fret should be placed 32 cm from the bridge.
To find the distance a fret should be place from the bridge
on a guitar, multiply the length of the string by ,
where n is the number of notes higher that the string’s
root note. Where should the fret be placed to produce the
E note that is one octave higher on the E string (12 notes
higher)?
Simplify.
Holt Algebra 2
8-6 Radical Expressions and Rational Exponents
Lesson Quiz: Part I
Find all real roots.
–5, 51. fourth roots of 625
2. fifth roots of –243 –3
Simplify each expression.
24. 4y2
5. Write (–216) in radical form and simplify.23
6. Write using rational exponents. 5321
3. 84 256y
= 36( )-2
3 2163 521
Holt Algebra 2
8-6 Radical Expressions and Rational Exponents
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?
Lesson Quiz: Part II
$2300.01