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[email protected] • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt1
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
§7.5 Denom§7.5 DenomRationalizeRationalize
[email protected] • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt2
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
[email protected] • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt3
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.
[email protected] • 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
[email protected] • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt5
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.
[email protected] • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt6
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
[email protected] • 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.
[email protected] • 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
[email protected] • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt9
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
[email protected] • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt10
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
( )
[email protected] • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt11
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
[email protected] • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt12
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
[email protected] • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt13
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
[email protected] • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt14
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.
[email protected] • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt15
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
[email protected] • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt16
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.
[email protected] • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt17
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
[email protected] • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt18
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
[email protected] • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt19
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
[email protected] • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt20
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
[email protected] • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt21
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
[email protected] • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt22
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
[email protected] • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt23
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
[email protected] • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt24
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
[email protected] • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt25
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)
[email protected] • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt26
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
[email protected] • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt27
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
AppendiAppendixx
–
srsrsr 22
[email protected] • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt28
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
[email protected] • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt29
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
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4
5
-10 -8 -6 -4 -2 0 2 4 6 8 10
M55_§JBerland_Graphs_0806.xls
x
y