§7.3 Radical Products Chabot Mathematics Bruce Mayer, PE

Chabot Mathematics

§7.3 Radical

Products

Bruce Mayer, PE

Licensed Electrical & Mechanical Engineer

BMayer@ChabotCollege.edu

Bruce Mayer, PE

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

Chabot College Mathematics

1

Review § 7.2

MTH 55

 Any QUESTIONS About

• §7.2 → Rational Exponents

 Any QUESTIONS About HomeWork

• §7.2 → HW-25


2

Bruce Mayer, PE


Multiplying Radical Expressions

 Note That:

4



9 2 3 6.

4 9 36



6.

 This example suggests the following.


3

Bruce Mayer, PE


Product Rule for Radicals

 n a a and n n b b , , n a

 n b

 n 

.

 That is, The product of two n th roots is the n th root of the product of the two radicands.


4

Bruce Mayer, PE


Derive Product Rule for Rads

 Rational exponents can be used to derive the Product Rule for

Radicals: n a

 n b

 a

1/ n  b

1/ n  a b

 1/ n  n a b .


5 n a

 n b

 n 

.

Bruce Mayer, PE


Example



Product Rule

 Multiply a) 5



6

 SOLUTION a) 5



6



5 6 30.

b) 7

 3

9 b) 7

 3

9

 3   3

63 c) 4 x

3



4

5 z c) 4 x

3



4

5 z



4 x

3

5 z

4

5

3 x z


6

Bruce Mayer, PE


Caveat Product Rule

 CAUTION

 The product rule for radicals applies only when radicals have the SAME index: n a

 m b

 nm a b .

Bruce Mayer, PE



7

Example



Product Rule

 Find the product and write the answer in simplest form. Assume all variables represent nonnegative values.

a) 4 2



4 8 b)

7 y

5 

7 y

9

 SOLN a) 4 2



4 8



4 2 8 b)

7 y

5 

7 y

9 

7 y

5  y

9


8



4 16



2



7 y

14

 y

2

Bruce Mayer, PE


Example



Product Rule

 Multiply  SOLUTION


9

Bruce Mayer, PE


Simplifying by Factoring

 The number p is a perfect square if there exists a rational number q for which q 2 = p . We say that p is a perfect n th power if q n = p for some rational number q .

 The product rule allows us to ab ab contains a factor that is a perfect n th power


10

Bruce Mayer, PE


Simplify by Product Rule

 Use The Product Rule in

REVERSE to Facilitate the

Simplification process n ab

 n a

 n b real numbers


11

Bruce Mayer, PE


Simplify a Radical Expression with Index n by Factoring

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

2. Take the n th root of each factor

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


12

Bruce Mayer, PE


Nix 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


13

Bruce Mayer, PE


Example



Simplify by Factoring

 Simplify by factoring a. 300 b. 8 m n

3 c. 54 s

4

 SOLUTION a. 300



100



3



100



3



10 3

100 is the largest perfectsquare factor of 300.


14

Bruce Mayer, PE


Example




 SOLUTION: b. 8 m n b. 8

4 m n



4 2 m

4  n

3 c. 54 s

4



4 m

4 



2 m

2

2 n

2 n

3 c. 54 s

4  3

27 2 s

3 s

27 s 3 is the largest perfect third-power factor.

 3

27 s




15



3 s

3

2 s

2 s

Bruce Mayer, PE


WhiteBoard 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*g 0.5

*SL 1.67

*h −1.17 (Alexander)


16

Bruce Mayer, PE


All Done for Today

Running

Dinosaur


17

Bruce Mayer, PE


Example




 Simplify  SOLUTION

Cannot be simplified further.


18

Bruce Mayer, PE


Chabot Mathematics

Appendix

r

2  s

2 

 r

 s

 r

 s



Bruce Mayer, PE

Licensed Electrical & Mechanical Engineer

BMayer@ChabotCollege.edu

–


19

Bruce Mayer, PE



20

Bruce Mayer, PE


Graph y = |x|

6

 Make T-table

y = |x | x

5

6

3

4

-1

0

1

2

-6

-5

-4

-3

-2

5

6

3

4

1

2

1

0

4

3

6

5

2

5

4

3

2

1

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

0

-1

0

-2

-3

-4

-5


21 y

1 2 3

-6 file =XY_Plot_0211.xls

Bruce Mayer, PE


4 5 x

6

-3 -2

2

1

-1

0

0

-1

-2

M55_§JBerland_Graphs_0806.xls

-3

5

4

3

1

4

5 y

2

-10 -8

3

3

2

-6

4

-4

5

1

-2

0

0

-1

-2

M55_§JBerland_Graphs_0806.xls

-3

-4

-5

2 4 6 8 x

10


22

Bruce Mayer, PE


§7.3 Radical Products Chabot Mathematics Bruce Mayer, PE

Chabot Mathematics

§7.3 Radical

Products

Running

Dinosaur

Chabot Mathematics

Appendix

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Products

Support

§7.3 Radical Products Chabot Mathematics Bruce Mayer, PE

Chabot Mathematics

§7.3 Radical

Products

Running

Dinosaur

Chabot Mathematics

Appendix

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