[email protected] MTH55_Lec-41_sec_7-3a_Radical_Product_Rule.ppt.ppt 1 Bruce Mayer, PE Chabot...

[email protected] • MTH55_Lec-41_sec_7-3a_Radical_Product_Rule.ppt.ppt1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§7.3 Radical§7.3 RadicalProductsProducts



Review §Review §

Any QUESTIONS About• §7.2 → Rational Exponents

Any QUESTIONS About HomeWork• §7.2 → HW-31

7.2 MTH 55



Multiplying Radical ExpressionsMultiplying Radical Expressions

Note That:

4 9 2 3 6.

4 9 36 6. This example suggests the

following.



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.



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



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.



Caveat Product RuleCaveat Product Rule

CAUTIONCAUTION The product rule for radicals applies

only when radicals have the SAME index:




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




Multiply SOLUTION



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



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



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.



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



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.



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.



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)



All Done for TodayAll Done for Today

RunningDinosaur



Example Example Simplify by Simplify by FactoringFactoring Simplify SOLUTION

Cannot be simplified further.



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

AppendiAppendixx

–

srsrsr 22



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



-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

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