bmayer@chabotcollege.edu mth55_lec-44_sec_7-5_rationalize_denoms.ppt 1 bruce mayer, pe chabot...

BMayer@ChabotCollege.edu • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

BMayer@ChabotCollege.edu

Chabot Mathematics

§7.5 Denom§7.5 DenomRationalizeRationalize

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Bruce Mayer, PE Chabot College Mathematics

Review §Review §

Any QUESTIONS About• §7.4 → Add, Subtract, Divide Radicals

Any QUESTIONS About HomeWork• §7.4 → HW-33

7.4 MTH 55

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Bruce Mayer, PE Chabot College Mathematics

Multiply RadicalsMultiply Radicals

Radical expressions often contain factors that have more than one term.

Multiplying such expressions is similar to finding products of polynomials.

Some products will yield like radical terms, which we can now combine.

BMayer@ChabotCollege.edu • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt4

Bruce Mayer, PE Chabot College Mathematics

Example Example Multiply Radicals Multiply Radicals

Find the Product for

3 6 5 7 7

SOLUTION

3 6 5 7 7 3 6 5 3 6 7 7

3 30 21 42

Use the distributive property.

Multiply Using ProductRule for Radicals

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Bruce Mayer, PE Chabot College Mathematics

Example Example Multiply Radicals Multiply Radicals

Find the Product for

SOLUTION (F.O.I.L.-like)

4 5 2 5 5 2 .

Use the product rule.

4 5 2 5 5 2

4 5 5 5 4 5 2 5 2 5 2 2

4 5 20 10 10 5 2

20 20 10 10 10

10 19 10

Use the distributive property.

Find the products.

Combine like radicals.

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Bruce Mayer, PE Chabot College Mathematics

Example Example Multiply Radicals Multiply Radicals

Find the Product for

SOLUTION

2

5 3

2 2

5 2 15 3 3

5 2 15 3

8 2 15

Simplify.

2

5 3 Use (a – b)2 = a2 – 2ab – b2

BMayer@ChabotCollege.edu • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt7

Bruce Mayer, PE Chabot College Mathematics

Example Example Multiply Radicals Multiply Radicals

Find the Product for

SOLUTION

8 3 8 3

228 3

61

8 3 8 3

64 3 Simplify.

Use (a + b)(a – b) = a2 – b2.

BMayer@ChabotCollege.edu • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt8

Bruce Mayer, PE Chabot College Mathematics

Example Example Multiply Radicals Multiply Radicals

Perform MultiTermMultiplication

3 23

a) 2( 7)

b) 2 3

c)

y

x x

m n m n

SOLUTION a)

2 14y

Using the distributive law

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Bruce Mayer, PE Chabot College Mathematics

Example Example Multiply Radicals Multiply Radicals

Perform MultiTermMultiplication

SOLUTION b)

3 33 233 2 6x x x

3 233 2 6x x x

F O I L

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Bruce Mayer, PE Chabot College Mathematics

Example Example Multiply Radicals Multiply Radicals

Perform MultiTermMultiplication

SOLUTION c)

2 2m m n m n n

m n

F O I L

( )

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Bruce Mayer, PE Chabot College Mathematics

Radical ConjugatesRadical Conjugates

In part (c) of the last example, notice that the inner and outer products in F.O.I.L. are opposites, the result, m – n, is not itself a radical expression. Pairs of radical terms like, are called conjugates.

and ,m n m n

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Bruce Mayer, PE Chabot College Mathematics

Mult. Radicals by Special ProdsMult. Radicals by Special Prods

Multiplication of expressions that contain radicals is very similar to the multiplication of polynomials

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Bruce Mayer, PE Chabot College Mathematics

Mult. Radicals by Special ProdsMult. Radicals by Special Prods

Compare F.O.I.L. and Square of a BiNomial-Sum

FOIL Method

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Bruce Mayer, PE Chabot College Mathematics

Rationalize DeNominatorRationalize DeNominator

When a radical expression appears in a denominator, it can be useful to find an equivalent expression in which the denominator NO LONGER contains a RADICAL. The procedure for finding such an expression is called rationalizing the denominator.

We carry this out by multiplying by 1 in either of two ways.

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Bruce Mayer, PE Chabot College Mathematics

Rationalize → Method-1Rationalize → Method-1

One way is to multiply by 1 under the radical to make the denominator of the radicand a perfect power.

EXAMPLE Rationalize Denom:

35

49

35 35

749

Multiplying by 1 under the radical

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Bruce Mayer, PE Chabot College Mathematics

Example Example Rationalize DeNom Rationalize DeNom

Rationalize DeNom:

SOLUTION

33

15

5

3

3 3

15

5

315

5

Since the index is 3, we need 3 identical factors in the denom.

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Bruce Mayer, PE Chabot College Mathematics

Rationalize → Method-2Rationalize → Method-2

Another way to rationalize a DeNom is to multiply by 1 outside the radical.

EXAMPLE Rationalize Denom:

5 3

3 3

x

x x

215

3

x

x 15

3

x

x

Multiplying by 1

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Bruce Mayer, PE Chabot College Mathematics

Example Example Rationalize DeNom Rationalize DeNom

Rationalize DeNom:

SOLN

23

3 33

3 2

8

y x y

x y

2 23 33 2 3 2

2 2

y x y x y

xy x

Need in DeNom Radical

3332 yx

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Bruce Mayer, PE Chabot College Mathematics

Example Example Rationalize DeNom Rationalize DeNom

Rationalize the denominator. Assume variables are >0 3

2

7

16x

SOLN Need in DeNom Radical 43x3

32

7

16x

3

3 2

7

16x

3

3 2

3

3

7

1 46

4

x

x

x

3

3 3

28

64

x

x

3 28

4

x

x

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Bruce Mayer, PE Chabot College Mathematics

Rationalize 2-Term Rad DeNomsRationalize 2-Term Rad DeNoms

Recall that the Difference-of-2Sqs Product results in the O & I terms in the FOIL Multiplication Adding to Zero

To Rationalize a DeNominator that contains two Radical Terms requires the use of Conjugates (which have a Diff-of-Sqs form) to remove the radicals from the Denom

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Bruce Mayer, PE Chabot College Mathematics

Rationalize 2-Term Rad DeNomsRationalize 2-Term Rad DeNoms

For Example to Rationalize the Denom of

Multiply the Numerator & Denominator by the CONJUGATE of the Original Denominator

2525

2454

25

25

23

2420

225

2420

225255

242022

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Bruce Mayer, PE Chabot College Mathematics

Example Example Rationalize DeNom Rationalize DeNom

Rationalize the denominator:

SOLUTION

5.

7 y

7

7

y

y

5 7

7 7

y

y y

25 7 5

7

y

y

5 5.

7 7y y

Multiplying by 1 using

the conjugate

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Bruce Mayer, PE Chabot College Mathematics

Example Example Rationalize DeNom Rationalize DeNom

Rationalize the denominator: 5 3

.3 5

3 5

3 5

Multiplying by 1 using

the conjugate

5 3

3 5

5 3

3 5

5 3 3 5

3 5 3 5

2 25 3 5 5 3 3 3 5

3 5

5 3 5 5 3 15

3 5

5 3 5 5 3 15

2

SOLUTION

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Bruce Mayer, PE Chabot College Mathematics

Rationalize NumeratorRationalize Numerator

To rationalize a numerator with more than one term, use the conjugate of the numerator

Example Rationalize numerator5 3

6

x

SOLUTION

5 3

6

x 5 3 5 3

6 5 3

x x

x

225 3

6 5 3

x

x

25 3

30 6 3

x

x

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Bruce Mayer, PE Chabot College Mathematics

WhiteBoard WorkWhiteBoard Work

Problems From §7.5 Exercise Set• 22, 38, 64, 74, 92, 128 → Derive φ

The Golden Ratioφ (phi)

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Bruce Mayer, PE Chabot College Mathematics

All Done for TodayAll Done for Today

L. Da VinciUsed The

Golden Ratio

Typo in Book for 1/GoldenRatio 15

2

15

2

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Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

BMayer@ChabotCollege.edu

Chabot Mathematics

AppendiAppendixx

–

srsrsr 22

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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

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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