[email protected] MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical

[email protected] • MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§7.2 Radical§7.2 RadicalFunctionsFunctions



Review §Review §

Any QUESTIONS About• §7.1 → Cube & nth Roots

Any QUESTIONS About HomeWork• §7.1 → HW-32

7.1 MTH 55



Rational ExponentsRational Exponents

Consider a1/2a1/2. If we still want to add exponents when multiplying, it must follow from the Exponent PRODUCT RULE that

a1/2a1/2 = a1/2 + 1/2, or a1

This suggests that a1/2 is a

square root of a.



Definition of Definition of aa1/1/nn

When a is NONnegative, n can be any natural number greater than 1. When a is negative, n must be odd.

Note that the denominator of the exponent becomes the index and the base becomes the radicand.

1/ = n na a1/ means .n na a



Evaluating Evaluating aa1/1/nn

Evaluate Each Expression

(a) 271/3 273 = 3

(b) 641/2 64 = 8

=

=

(c) –6251/4 6254 = –5= –

(d) (–625)1/4 –6254 is not a real number because the radicand,

–625, is negative and the index is even.

=



Caveat on RootsCaveat on Roots

CAUTIONCAUTION: Notice the difference between parts (c) and (d) in the last Example.

The radical in part (c) is the negative fourth root of a positive number, while the radical in part (d) is the principal fourth root of a negative number, which is NOT a real no.

(c) –6251/4 6254 = –5= –

(d) (–625)1/4 –6254 is not a real number because the radicand,

–625, is negative and the index is even.=



Radical FunctionsRadical Functions

Given PolyNomial, P, a RADICAL FUNCTION Takes this form:

Example Given f(x) = Then find f(3).

n Pxf

5 8,x

SOLUTION To find f(3), substitute 3 for x and simplify.

3 5 3 8f 15 8 7



Example Example Exponent to Radical Exponent to Radical

Write an equivalent expression using RADICAL notation

a) b) c)3

1

m

4442

1

2

18 3999 xxxx

5 25

12 zxyzxy

SOLUTION

33

1

mm a)

b)

c)

2

189x 5

12zxy



Example Example Radical to Exponent Radical to Exponent

Write an equivalent expression using EXPONENT notation

a) b)

SOLUTION

a) b)

512

4

2 55

x

y

x

y

3 4x 4

25

x

y

313 44 xx



Exponent ↔ IndexExponent ↔ IndexBase ↔ RadicandBase ↔ Radicand From the Previous Examples Notice:

33

1

mm The denominator of the exponent

becomes the index. The base becomes the radicand.

313 44 xx The index becomes the denominator of the exponent. The radicand becomes the base.



Definition of Definition of aamm//nn

For any natural numbers m and n (n not 1) and any real number a for which the radical exists,

/ means , or .m nm n mna a a

n a



Example Example aamm//nn Radicals Radicals

Rewrite as radicals, then simplify• a. 272/3 b. 2433/4 c. 95/2

SOLUTIONa. 2/3 1/3 227 (27 ) 23( 27) 23 9

b. 3/5 1/5 3243 (243 ) 35( 243) 33 27

c. 5/ 2 1/ 2 59 (9 ) 53 243 5( 9)



Example Example aamm//nn Exponents Exponents

Rewrite with rational exponents

SOLUTION

2 2 / 55b) 3 3xy xy

3 5 5 / 3a) x x



Definition of Definition of aa−−mm//nn

For any rational number m/n and any positive real number a the NEGATIVE rational exponent:

//

1 means .m n

m na

a

That is, am/n and a−m/n are reciprocals



Caveat on Negative ExponentsCaveat on Negative Exponents

A negative exponent does not indicate that the expression in which it appears is negative; i.e.;



Example Example Negative Exponents Negative Exponents

Rewrite with positive exponents, & simplify• a. 8−2/3 b. 9−3/2x1/5 c.

3/ 43

2

t

r

SOLUTION

3/ 2 1/ 5 1/ 53/ 21

b) 99

x x

1/ 51/ 51

27 27

xx

3/ 4 3/ 43 2c)

2 3

t r

r t

2 / 32 / 31

a) 88

21 1

42

23

1

8



Example Example Speed of Sound Speed of Sound

Many applications translate to radical equations.

For example, at a temperature of t degrees Fahrenheit, sound travels S feet per second According to the Formula



Example Example Speed of Sound Speed of Sound

During orchestra practice, the temperature of a room was 74 °F. How fast was the sound of the orchestra traveling through the room?

SOLUTION:Substitute 74 for t in the Formula and find an approximation using a calculator.

21.9 5 2457S t

721.9 5( 4 74) 2 5S

21.9 370 2457S

21.9 2827S

1164.4 ft/sec.S hr

mile 9.793

ft 5280

mile 1

hr 1

s 3600

s

ft 4.1164



WhiteBoard WorkWhiteBoard Work

Problems From §7.2 Exercise Set• 4, 10, 18, 32, 48, 54, 130

The MACH No.M



All Done for TodayAll Done for Today

Ernst MachFluid

Dynamicist

Born 8Feb1838 in Brno, Austria

Died 19Feb1916 (aged 78) in Munich, Germany



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

AppendiAppendixx

–

srsrsr 22



hr

mile 9.793

ft 5280

mile 1

hr 1

s 3600

s

ft 4.1164



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



-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

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[email protected] MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical