BMayer@ChabotCollege.edu MTH55_Lec-43_sec_7-4_Add_Sub_Divide_Radicals.ppt 1 Bruce Mayer, PE Chabot...

BMayer@ChabotCollege.edu • MTH55_Lec-43_sec_7-4_Add_Sub_Divide_Radicals.ppt1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

BMayer@ChabotCollege.edu

Chabot Mathematics

§7.4 +/§7.4 +/−/−/RadicalsRadicals

BMayer@ChabotCollege.edu • MTH55_Lec-43_sec_7-4_Add_Sub_Divide_Radicals.ppt2

Bruce Mayer, PE Chabot College Mathematics

Review §Review §

Any QUESTIONS About• §7.3 → Multiply Radicals

Any QUESTIONS About HomeWork• §7.3 → HW-26

7.3 MTH 55

BMayer@ChabotCollege.edu • MTH55_Lec-43_sec_7-4_Add_Sub_Divide_Radicals.ppt3

Bruce Mayer, PE Chabot College Mathematics

Add & Subtract RadicalsAdd & Subtract Radicals

Radical Expressions similar to those that follow can be simplified using the distributive property

These canNOT be combined

BMayer@ChabotCollege.edu • MTH55_Lec-43_sec_7-4_Add_Sub_Divide_Radicals.ppt4

Bruce Mayer, PE Chabot College Mathematics

CAVEAT Identical IndicesCAVEAT Identical Indices

CAUTIONCAUTION Only Radical Expressions with the

SAME INDEX and SAME RADICAND may be combined

Expressions such as Those below CanNOT be simplified by Combining Terms

BMayer@ChabotCollege.edu • MTH55_Lec-43_sec_7-4_Add_Sub_Divide_Radicals.ppt5

Bruce Mayer, PE Chabot College Mathematics

Example Example Simplify by Add/Sub Simplify by Add/Sub

Simplify by Adding or Subtractinga) b)

SOLN → chk 1st Indices & Radicands a) b)

BMayer@ChabotCollege.edu • MTH55_Lec-43_sec_7-4_Add_Sub_Divide_Radicals.ppt6

Bruce Mayer, PE Chabot College Mathematics

Example Example Simplify by Add/Sub Simplify by Add/Sub

Simplify by Adding or Subtractinga) b)

SOLN → chk 1st Indices & Radicandsa)

28 7 63 112 28

4 7 7 Factor 28.28 7

2 7 7 3 7 Combine like radicals.

Simplify Radical

b) 63 112 28 9 7 16 7 4 7

3 7 4 7 2 7

3 7

BMayer@ChabotCollege.edu • MTH55_Lec-43_sec_7-4_Add_Sub_Divide_Radicals.ppt7

Bruce Mayer, PE Chabot College Mathematics

Example Example Simplify by Add/Sub Simplify by Add/Sub

Simplify by Adding or Subtractinga) b)

SOLN → chk 1st Indices & Radicandsa)

b)

14 5 11 2 11 5 5

3 37 5 12 5x x 35 5xCombine the like radicals by subtracting the coefficients and keeping the radical.

14 5 11 2 11 5 5

(14 5) 5 (2 1) 11

9 5 11

Regroup terms.

3 37 5 12 5x x

BMayer@ChabotCollege.edu • MTH55_Lec-43_sec_7-4_Add_Sub_Divide_Radicals.ppt8

Bruce Mayer, PE Chabot College Mathematics

Example Example Simplify by Add/Sub Simplify by Add/Sub

Simplify by Adding or Subtracting

SOLN → chk 1st Indices & Radicands

BMayer@ChabotCollege.edu • MTH55_Lec-43_sec_7-4_Add_Sub_Divide_Radicals.ppt9

Bruce Mayer, PE Chabot College Mathematics

Example Example Simplify by Add/Sub Simplify by Add/Sub

Simplify by Adding or Subtracting (assume a, b > 0)

SOLN → Note that these are CUBE Rts

BMayer@ChabotCollege.edu • MTH55_Lec-43_sec_7-4_Add_Sub_Divide_Radicals.ppt10

Bruce Mayer, PE Chabot College Mathematics

Radical Expression DivisionRadical Expression Division

Just as the root of a product can be expressed as the product of two roots, the root of a quotient can be expressed as the quotient of two roots.

BMayer@ChabotCollege.edu • MTH55_Lec-43_sec_7-4_Add_Sub_Divide_Radicals.ppt11

Bruce Mayer, PE Chabot College Mathematics

Quotient Rule for RadicalsQuotient Rule for Radicals

For any real numbers andwith b > 0

Remember that an nth root is simplified when its radicand has no factors that are perfect nth pwrs • Recall also that we assume that no radicands

represent negative quantities raised to an even power

BMayer@ChabotCollege.edu • MTH55_Lec-43_sec_7-4_Add_Sub_Divide_Radicals.ppt12

Bruce Mayer, PE Chabot College Mathematics

Example Example Simplify by Quotient Simplify by Quotient

Simplify by taking roots of the numerator and denominator:

a) b)

SOLN a)

27

x Taking the square roots of the

numerator & denominator

BMayer@ChabotCollege.edu • MTH55_Lec-43_sec_7-4_Add_Sub_Divide_Radicals.ppt13

Bruce Mayer, PE Chabot College Mathematics

Example Example Simplify by Quotient Simplify by Quotient

Simplify by taking roots of the numerator & denominator:

b)

SOLN b) →

3 3

3 3

27 2 m m

n

3 3 3

3 3

27 2m m

n

33 2m m

n

BMayer@ChabotCollege.edu • MTH55_Lec-43_sec_7-4_Add_Sub_Divide_Radicals.ppt14

Bruce Mayer, PE Chabot College Mathematics

WhiteBoard WorkWhiteBoard Work

Problems From §7.4 Exercise Set• 10, 26, 28, 40, 56, 60, 84

Recall Σ & Δ of Two Cubes• DIFFERENCE of Cubes:

• a³ − b³ = (a − b)(a² + ab + b²)

• SUM of Cubes:

• a³ + b³ = (a + b)(a² − ab + b²)

BMayer@ChabotCollege.edu • MTH55_Lec-43_sec_7-4_Add_Sub_Divide_Radicals.ppt15

Bruce Mayer, PE Chabot College Mathematics

All Done for TodayAll Done for Today

AlbertEinstein

Born March 14, 1879 in Ulm, Württemberg, Germany

Died April 18, 1955 (age 76) in Princeton, New Jersey, USA

BMayer@ChabotCollege.edu • MTH55_Lec-43_sec_7-4_Add_Sub_Divide_Radicals.ppt16

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-43_sec_7-4_Add_Sub_Divide_Radicals.ppt17

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-43_sec_7-4_Add_Sub_Divide_Radicals.ppt18

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