Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

[email protected] • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§7.1 Cube§7.1 Cube& n& nthth Roots Roots



Review §Review §

Any QUESTIONS About• §7.1 → Square-Roots and Radical

Expessions

Any QUESTIONS About HomeWork• §7.1 → HW-30

7.1 MTH 55



Cube RootCube Root

The CUBE root, c, of a Number a is written as:

The number c is the cube root of a, if the third power of c is a; that is; if c3 = a, then



Example Example Cube Root of No.s Cube Root of No.s

Find Cube Rootsa) b) c)3 0.008 3 27

643 2197

SOLUTION• a) As 0.2·0.2·0.2 = 0.008

• b) 1321973 As (−13)(−13)(−13) = −2197

• c) As 33 = 27 and 43 = 64,so (3/4)3 = 27/64



Cube Root FunctionsCube Root Functions

Since EVERY Real Number has a Cube Root Define the Cube Root Function:

3 xxf The Graph

Reveals• Domain =

{all Real numbers}

• Range ={all Real numbers}

-5

-4

-3

-2

-1

0

1

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5

-10 -8 -6 -4 -2 0 2 4 6 8 10

M55_§JBerland_Graphs_0806.xls

x

y

x

y

3 xxfy



Evaluate Cube Root FunctionsEvaluate Cube Root Functions

Evaluate Cube Root Functionsa)

b)

731923 uyyu

173773 vzzv

SOLUTION (using calculator)a)

b)

0255127

1914619732733

33

.

u

344482

3711937177173

33

.

v



Simplify Cube RootsSimplify Cube Roots

For any Real Number, a aa 3 3

Use this property to simplify Cube Root Expressions.

For EXAMPLE Simplify

SOLUTION

because (–3x)(–3x)(–3x) = –27x3



nnthth Roots Roots

nth root: The number c is an nth root of a number a if cn = a.

The fourth root of a number a is the number c for which c4 = a. We write for the nth root. The number n is called the index (plural, indices). When the index is 2 (for a Square Root), the Index is ommitted.

n a



Odd & Even Odd & Even nnthth Roots → Roots →

When the index number, n, is ODD the root itself is also called ODD• A Cube-Root (n = 3) is Odd. Other Odd

roots share the properties of Cube-Roots– the most important property of ODD roots is

that we can take the ODD-Root of any Real Number – positive or NEGATIVE

– Domain of Odd Roots = (−, +)

– Range of Odd Roots =(−, +)

n a



Example Example nnthth Roots of No.s Roots of No.s

Find ODD Rootsa) b) c)

SOLUTION• a) Since 35 = 243

• b)

5 243

As (−3)(−3)(−3)(−3)(−3) = −243

• c)When the index equals the exponent under the radical we recover the Base

5 243 11 11m

32435

32435

mm 11 11



Odd & Even Odd & Even nnthth Roots → Roots →

When the index number, n, is EVEN the root itself is also called EVEN• A Sq-Root (n = 2) is Even. Other Even

roots share the properties of Sq-Roots– The most important property of EVEN roots is

that we canNOT take the EVEN-Root of a NEGATIVE number.

– Domain of Even Roots = {x|x ≥ 0}

– Range of Even Roots = {y|y ≥ 0}

n a



Example Example nnthth Roots of No.s Roots of No.s

Find EVEN Rootsa) b) c)

SOLUTION• a) Since 34 = 81

• b)

4 81

Even Root is Not a Real No.

• c)Use absolute-value notation since m could represent a negative number

4 81 4 416m

3814

4 81

mm 2164 4



Simplifying Simplifying nnthth Roots Roots

n a

Even

Positive Positive a

Negative Not a real number

|a|

Odd

Positive Positive a

Negative Negative a

n a nn a



Example Example Radical Radical ExpressionsExpressions Find nth Roots

a) b) c)

SOLUTION• a)

• b)

4 42 7u

• c)

5 5113 v 6 613

EVEN is if as 774 4 naauu n n

ODD is if as 1131135 5 naavv n n

EVEN is if as 1313136 6 naan n



WhiteBoard WorkWhiteBoard Work

Problems From §7.1 Exercise Set• 50, 74, 84, 88, 98, 102

Principalnth Root



All Done for TodayAll Done for Today

SkidMarkAnalysis

Skid Distances



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

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-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

file =XY_Plot_0211.xls



y

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-18 -16 -14 -12 -10 -8 -6 -4 -2 0 2

M55_§JBerland_Graphs_0806.xls

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xy

Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

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