Algebra 2 unit 6.1

UNIT 6.1 ROOTS AND UNIT 6.1 ROOTS AND RADICAL EXPRESSIONSRADICAL EXPRESSIONS

Warm Up

Simplify each expression.

16,807

121

729

1. 73 • 72

3. (32)3

2. 118

116

4.

5.

75

207

2 357

5 3

Rewrite radical expressions by using rational exponents.

Simplify and evaluate radical expressions and expressions containing rational exponents.

Objectives

indexrational exponent

Vocabulary

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.

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

When a radical sign shows no index, it represents a square root.

Reading Math

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.


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.

The properties of square roots in Lesson 1-3 also apply to nth roots.

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!

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

Example 2B: Simplifying Radical Expressions

Quotient Property.

Product Property.

Simplify the numerator.

Rationalize the numerator.

Simplify.

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

Check It Out! Example 2b

Quotient Property.

Product Property.

Rationalize the numerator.

Simplify.


84

4 3

x

4227

3x

Check It Out! Example 2c

3 37 2x xg

Product Property of Roots.

Simplify.


x3

3 9x

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

The denominator of a rational exponent becomes the index of the radical.

Writing Math

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.


–8

Evaluate the power.

Evaluate the root.

( )−3

5 32

−5 32,768

Method 1 Evaluate the root first.

(4)1


6413

4

Evaluate the root.

Evaluate the power.


Write the expression in radical form, and simplify.


Write will a radical.

4

Evaluate the power.

Evaluate the root.

( )13 64 ( )13 64

3 64

Method 1 Evaluatethe root first.

(2)5


452

32

Evaluate the root.

Evaluate the power.


Write the expression in radical form, and simplify.

Method 2 Evaluatethe power first.


32

Evaluate the power.

Evaluate the root.

( )52 4 ( )52 4

2 1024

Method 1 Evaluatethe root first.

(5)3


62534

125

Evaluate the root.

Evaluate the power.

Check It Out! Example 3c Write the expression in radical form, and simplify.



125

Evaluate the power.

Evaluate the root.

( )34 625 ( )34 625

4 244,140,625

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

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. 512

c.

24 5 =

mmn na a

Rational exponents have the same properties as integer exponents (See Lesson 1-5)

Example 5A: Simplifying Expressions with Rational Exponents

Product of Powers.


Simplify.

Evaluate the Power.

72

49

Check Enter the expression in a graphing calculator.

Example 5B: Simplifying Expressions with Rational Exponents

Quotient of Powers.


Simplify.

Negative Exponent Property.

1 4

Evaluate the power.

Example 5B Continued


Product of Powers.


Simplify.

Evaluate the Power.6




(–8)–13


1 –8

13

1 2

–


Evaluate the Power.


Quotient of Powers.


Simplify.

Evaluate the power.

52

Check It Out! Example 5c

25


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?

Example 6 ContinuedSubstitute 800 for t.

Simplify.


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.


Use 64 cm for the length of the string, and substitute 12 for n.

Check It Out! Music Application

= 64(2–1)

Simplify.= 32The 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.

Lesson Quiz: Part I


–5, 51. fourth roots of 625

2. fifth roots of –243 –3


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

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

All rights belong to their respective owners.Copyright Disclaimer Under Section 107 of the Copyright Act 1976, allowance is made for "fair use" for purposes such as criticism, comment, news reporting, TEACHING, scholarship, and research. Fair use is a use permitted by copyright statute that might otherwise be infringing. Non-profit, EDUCATIONAL or personal use tips the balance in favor of fair use.

Algebra 2 unit 6.1

Education

Transcript of Algebra 2 unit 6.1