BMayer@ChabotCollege.edu • MTH55_Lec-41_sec_7-3a_Radical_Product_Rule.ppt.ppt1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

BMayer@ChabotCollege.edu

Chabot Mathematics

§7.3 Radical§7.3 RadicalProductsProducts

BMayer@ChabotCollege.edu • MTH55_Lec-41_sec_7-3a_Radical_Product_Rule.ppt.ppt2

Bruce Mayer, PE Chabot College Mathematics

Review §Review §

Any QUESTIONS About• §7.2 → Rational Exponents

Any QUESTIONS About HomeWork• §7.2 → HW-31

7.2 MTH 55

BMayer@ChabotCollege.edu • MTH55_Lec-41_sec_7-3a_Radical_Product_Rule.ppt.ppt3

Bruce Mayer, PE Chabot College Mathematics

Multiplying Radical ExpressionsMultiplying Radical Expressions

Note That:

4 9 2 3 6.

4 9 36 6. This example suggests the

following.

BMayer@ChabotCollege.edu • MTH55_Lec-41_sec_7-3a_Radical_Product_Rule.ppt.ppt4

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.

BMayer@ChabotCollege.edu • MTH55_Lec-41_sec_7-3a_Radical_Product_Rule.ppt.ppt5

Bruce Mayer, PE Chabot College Mathematics

Derive Product Rule for RadsDerive Product Rule for Rads

Rational exponents can be used to derive the Product Rule for Radicals:

1/1/ 1/ .nn nn n na b a b a b a b

BMayer@ChabotCollege.edu • MTH55_Lec-41_sec_7-3a_Radical_Product_Rule.ppt.ppt6

Bruce Mayer, PE Chabot College Mathematics

Example Example Product Rule Product Rule

Multiply

a) 5 6

44 5c)

3

x

z

3 3b) 7 9

SOLUTION

3 3 3 3b) 7 9 7 9 63

44 4 45 5 5c)

3 3 3

x x x

z z z

a) 5 6 5 6 30.

BMayer@ChabotCollege.edu • MTH55_Lec-41_sec_7-3a_Radical_Product_Rule.ppt.ppt7

Bruce Mayer, PE Chabot College Mathematics

Caveat Product RuleCaveat Product Rule

CAUTIONCAUTION The product rule for radicals applies

only when radicals have the SAME index:

BMayer@ChabotCollege.edu • MTH55_Lec-41_sec_7-3a_Radical_Product_Rule.ppt.ppt8

Bruce Mayer, PE Chabot College Mathematics

Example Example Product Rule Product Rule

Find the product and write the answer in simplest form. Assume all variables represent nonnegative values.a) b)4 42 8 5 97 7y y

SOLNa) b)4 42 8 4 2 8

4 16

2

5 97 7y y 5 97 y y

147 y

2y

BMayer@ChabotCollege.edu • MTH55_Lec-41_sec_7-3a_Radical_Product_Rule.ppt.ppt9

Bruce Mayer, PE Chabot College Mathematics

Example Example Product Rule Product Rule

Multiply SOLUTION

BMayer@ChabotCollege.edu • MTH55_Lec-41_sec_7-3a_Radical_Product_Rule.ppt.ppt10

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

BMayer@ChabotCollege.edu • MTH55_Lec-41_sec_7-3a_Radical_Product_Rule.ppt.ppt11

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

BMayer@ChabotCollege.edu • MTH55_Lec-41_sec_7-3a_Radical_Product_Rule.ppt.ppt12

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.

BMayer@ChabotCollege.edu • MTH55_Lec-41_sec_7-3a_Radical_Product_Rule.ppt.ppt13

Bruce Mayer, PE Chabot College Mathematics

Nix Negative RadicandsNix Negative Radicands

It is often safe to assume that a radicand does not represent a negative number when the radicand is raised to an even power

To Clarify the Essence of Radical Simplification We will make this assumption• i.e., do NOT Need AbsVal bars

BMayer@ChabotCollege.edu • MTH55_Lec-41_sec_7-3a_Radical_Product_Rule.ppt.ppt14

Bruce Mayer, PE Chabot College Mathematics

Example Example Simplify by Factoring Simplify by Factoring

Simplify by factoring

a. 300 4b. 8m n 3 4c. 54s

SOLUTION

a. 300 3100

100 3

10 3

100 is the largest perfect-square factor of 300.

BMayer@ChabotCollege.edu • MTH55_Lec-41_sec_7-3a_Radical_Product_Rule.ppt.ppt15

Bruce Mayer, PE Chabot College Mathematics

Example Example Simplify by Factoring Simplify by Factoring

SOLUTION: 4b. 8m n 3 4c. 54s

4 4b. 8 4 2m n m n

44 2m n 22 2m n

3 34 3c. 54 27 2s s s

3 3 327 2s s 33 2s s

27s3 is the largest perfect third-power factor.

BMayer@ChabotCollege.edu • MTH55_Lec-41_sec_7-3a_Radical_Product_Rule.ppt.ppt16

Bruce Mayer, PE Chabot College Mathematics

WhiteBoard WorkWhiteBoard Work

Problems From §7.3 Exercise Set• 14, 18, 20, 32,

38, 56, 96

Estimating Dinosaur Speed (h ≡ hip hgt) v = [gh(SL/1.8h)2.56]0.5 (Thulborn) v = 0.25*g0.5*SL1.67*h−1.17 (Alexander)

BMayer@ChabotCollege.edu • MTH55_Lec-41_sec_7-3a_Radical_Product_Rule.ppt.ppt17

Bruce Mayer, PE Chabot College Mathematics

All Done for TodayAll Done for Today

RunningDinosaur

BMayer@ChabotCollege.edu • MTH55_Lec-41_sec_7-3a_Radical_Product_Rule.ppt.ppt18

Bruce Mayer, PE Chabot College Mathematics

Example Example Simplify by Simplify by FactoringFactoring Simplify SOLUTION

Cannot be simplified further.

BMayer@ChabotCollege.edu • MTH55_Lec-41_sec_7-3a_Radical_Product_Rule.ppt.ppt19

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

BMayer@ChabotCollege.edu

Chabot Mathematics

AppendiAppendixx

–

srsrsr 22

BMayer@ChabotCollege.edu • MTH55_Lec-41_sec_7-3a_Radical_Product_Rule.ppt.ppt20

Bruce Mayer, PE Chabot College Mathematics

BMayer@ChabotCollege.edu • MTH55_Lec-41_sec_7-3a_Radical_Product_Rule.ppt.ppt21

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

BMayer@ChabotCollege.edu • MTH55_Lec-41_sec_7-3a_Radical_Product_Rule.ppt.ppt22

Bruce Mayer, PE Chabot College Mathematics

-3

-2

-1

0

1

2

3

4

5

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

M55_§JBerland_Graphs_0806.xls -5

-4

-3

-2

-1

0

1

2

3

4

5

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

M55_§JBerland_Graphs_0806.xls

x

y