[email protected] mth55_lec-42_sec_7-3b_factor_radicals.ppt 1 bruce mayer, pe chabot college...

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[email protected] • MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer [email protected] Chabot Mathematics §7.3 Factor §7.3 Factor Radicals Radicals

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Page 1: BMayer@ChabotCollege.edu MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical

[email protected] • MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§7.3 Factor§7.3 FactorRadicalsRadicals

Page 2: BMayer@ChabotCollege.edu MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical

[email protected] • MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt2

Bruce Mayer, PE Chabot College Mathematics

Review §Review §

Any QUESTIONS About• §7.3 → Multiply Radicals

Any QUESTIONS About HomeWork• §7.3 → HW-32

7.3 MTH 55

Page 3: BMayer@ChabotCollege.edu MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical

[email protected] • MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt3

Bruce Mayer, PE Chabot College Mathematics

Product Rule for RadicalsProduct Rule for Radicals

For any real numbers and

That is, The product of two nth roots is the nth root of the product of the two radicands.

Page 4: BMayer@ChabotCollege.edu MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical

[email protected] • MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt4

Bruce Mayer, PE Chabot College Mathematics

Simplifying by FactoringSimplifying by Factoring

The number p is a perfect square if there exists a rational number q for which q2 = p. We say that p is a perfect nth power if qn = p for some rational number q.

The product rule allows us to simplify whenever ab contains a factor that is a perfect nth power

n ab

Page 5: BMayer@ChabotCollege.edu MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical

[email protected] • MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt5

Bruce Mayer, PE Chabot College Mathematics

Simplify by Product RuleSimplify by Product Rule

Use The Product Rule in REVERSE to Facilitate the Simplification process

• Note that and must both be real numbers

n n nab a b

Page 6: BMayer@ChabotCollege.edu MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical

[email protected] • MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt6

Bruce Mayer, PE Chabot College Mathematics

Simplify a Radical Expression Simplify a Radical Expression with Index with Index nn by Factoring by Factoring

1. Express the radicand as a product in which one factor is the largest perfect nth power possible.

2. Take the nth root of each factor

3. Simplification is complete when no radicand has a factor that is a perfect nth power.

Page 7: BMayer@ChabotCollege.edu MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical

[email protected] • MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt7

Bruce Mayer, PE Chabot College Mathematics

Example Example Simplify by Factoring Simplify by Factoring

Simplify by factoring (assume x > 0)a) b)

SOLUTION → Match INDICES a) b)

Page 8: BMayer@ChabotCollege.edu MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical

[email protected] • MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt8

Bruce Mayer, PE Chabot College Mathematics

Example Example Simplify by Factoring Simplify by Factoring

Simplify by factoring (assume x > 0)a) b)

SOLN a)NoteThat theINDEXis 3

Page 9: BMayer@ChabotCollege.edu MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical

[email protected] • MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt9

Bruce Mayer, PE Chabot College Mathematics

Example Example Simplify by Factoring Simplify by Factoring

Simplify by factoring (assume x > 0)a) b)

SOLN b) Note INDEX of 5

Page 10: BMayer@ChabotCollege.edu MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical

[email protected] • MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt10

Bruce Mayer, PE Chabot College Mathematics

Example Example Simplify by Factoring Simplify by Factoring

Simplify by factoring (assume x, y > 0)a) b)

SOLN a) Note INDEX of 2

4 15

2 15

Page 11: BMayer@ChabotCollege.edu MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical

[email protected] • MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt11

Bruce Mayer, PE Chabot College Mathematics

Example Example Simplify by Factoring Simplify by Factoring

Simplify by factoring (assume x, y > 0)a) b)

SOLN b) Note INDEX of 3

6 93 27 3 x x y

3 6 933 3 27 3x y x

2 33 3 3x y x

Page 12: BMayer@ChabotCollege.edu MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical

[email protected] • MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt12

Bruce Mayer, PE Chabot College Mathematics

Example Example Simplify Simplify

Simplify by factoring (assume w, z > 0)

SOLN: First perform Distribution• Note that all INDICES are common at 5

5 965 75 64 428 zwwzzw

5 965 645 75 64

5 965 75 64

4828

428

zwzwwzzw

zwwzzw

Page 13: BMayer@ChabotCollege.edu MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical

[email protected] • MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt13

Bruce Mayer, PE Chabot College Mathematics

Example Example Simplify Simplify

SOLN: Use Radical Product Rule

SOLN: Use Commutative Property of Multiplication

5 96645 764

5 965 645 75 64

4828

4828

zwzwwzzw

zwzwwzzw

5 96645 764

5 96645 764

4828

4828

zzwwzzww

zwzwwzzw

Page 14: BMayer@ChabotCollege.edu MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical

[email protected] • MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt14

Bruce Mayer, PE Chabot College Mathematics

Example Example Simplify Simplify

SOLN: Exponent Product Rule

5 151055 1354

5 96645 7614

5 96645 764

22

3216

4828

zwzw

zwzw

zzwwzzww

SOLN: Next use Exponent POWER rule to expose as many bases as possible to the Power of 5; the Radical Index

Page 15: BMayer@ChabotCollege.edu MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical

[email protected] • MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt15

Bruce Mayer, PE Chabot College Mathematics

Example Example Simplify Simplify

SOLN: Power-to-Power Exponent Rule

5 535255 3525

5 151055 1354

216

22

zwzzw

zwzw

SOLN: Next Radical Product Rule

5 535 525 55 525 55 3

5 535255 3525

216

216

zwzwz

zwzzw

Page 16: BMayer@ChabotCollege.edu MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical

[email protected] • MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt16

Bruce Mayer, PE Chabot College Mathematics

Example Example Simplify Simplify

SOLN: Perform 5th Root Operations

SOLN: Finally Factor GCF = wz2

5 42323225 3

5 535 525 55 525 55 3

162216

216

zwzzwzwzwz

zwzwz

5 32

5 3232

162

162

zwzwz

zwzzw

ANS

Page 17: BMayer@ChabotCollege.edu MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical

[email protected] • MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt17

Bruce Mayer, PE Chabot College Mathematics

WhiteBoard WorkWhiteBoard Work

Problems From §7.3 Exercise Set• 76, 80, 82, 92, 98

AdultCardiacIndex =2.8-3.4

Page 18: BMayer@ChabotCollege.edu MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical

[email protected] • MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt18

Bruce Mayer, PE Chabot College Mathematics

All Done for TodayAll Done for Today

ExponentRules are

NOTAlgebraic

Page 19: BMayer@ChabotCollege.edu MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical

[email protected] • MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt19

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

AppendiAppendixx

srsrsr 22

Page 20: BMayer@ChabotCollege.edu MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical

[email protected] • MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt20

Bruce Mayer, PE Chabot College Mathematics

Graph Graph yy = | = |xx||

Make T-tablex y = |x |

-6 6-5 5-4 4-3 3-2 2-1 10 01 12 23 34 45 56 6

x

y

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

file =XY_Plot_0211.xls

Page 21: BMayer@ChabotCollege.edu MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical

[email protected] • MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt21

Bruce Mayer, PE Chabot College Mathematics

-3

-2

-1

0

1

2

3

4

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-3 -2 -1 0 1 2 3 4 5

M55_§JBerland_Graphs_0806.xls -5

-4

-3

-2

-1

0

1

2

3

4

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-10 -8 -6 -4 -2 0 2 4 6 8 10

M55_§JBerland_Graphs_0806.xls

x

y