[email protected] MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt 1 Bruce Mayer, PE Chabot...

[email protected] • MTH55_Lec-40_sec_7-2b_Rational_Exponents.ppt1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§7.2 Rational§7.2 RationalExponentsExponents



Review §Review §

Any QUESTIONS About• §7.2 → Radical Functions

Any QUESTIONS About HomeWork• §7.2 → HW-31

7.2 MTH 55



Laws of ExponentsLaws of Exponents

For any real number a, any real number b > 0, and any rational exponents m & n.1.

2.

3.

4.

5.

m n m na a a m

m nn

aa

a

nm m na a

m m mab a bn n

na a

b b

In multiplying, we can add exponents if the bases are the same.In dividing, we can subtract exponents if the bases are the same.

To raise a power to a power, we can multiply the exponents.

To raise a product to a power, we can raise each factor to the power.

To raise a quotient to a power, raise both the numerator & denominator to the power.



Example Example Laws of Exponents Laws of Exponents

Use the rules of exponents to simplify. Write the answer with only positive exponents

5/ 6

1/ 6

y

y

SOLUTIONUse the quotient for exponents. (Subtract the exponents.)

Rewrite the subtraction as addition.

Add the exponents.

5/ 6

1/ 6

y

y 5/ 6 ( 1/ 6)y

5/ 6 1/ 6y

y




Use the Laws of Exponents to Simplify

2 / 5 1/ 5a. 7 71/ 2

1/ 4b.

m

m 3/ 41/ 2 1/ 3c. x y

SOLUTION

1/ 21/ 2 1/ 4 2 / 4 1/ 4 1/ 4

1/ 4b)

mm m m

m

2 / 5 1/ 5 2 / 5 1/ 5 3/ 5a) 7 7 7 7




Use the Laws of Exponents to Simplify

3/ 41/ 2 1/ 3c. x y SOLUTION

3/ 83/ 8 1/ 4

1/ 4x

x yy

3/ 41/ 2 1/ 3 (1/ 2)(3 / 4) ( 1/ 3)(3 / 4)c) x y x y




Write with only positive exponents. Assume that all variables are ≥ 0

Power-to-Power rule

m1/4 n–6

m–8 n2/3

–3/4 =( m–8)–3/4 (n2/3)–3/4

(m1/4)–3/4 (n–6)–3/4

=m6 n–1/2

m–3/16 n9/2

= m–3/16 – 6 n9/2 – (–1/2) Quotient rule

= m–99/16 n5

Definition of

Negative exponent=

m99/16

n5

Product to Power




Write with only positive exponents. All variables represent positive numbers

x3/5(x–1/2 – x3/4) = x3/5 · x–1/2 – x3/5 · x3/4 Distributive property

= x3/5 + (–1/2) – x3/5 + 3/4 Product rule

= x1/10 – x27/20

Do not make the common mistake of multiplying exponents in the first step.



Simplifying Radical ExpressionsSimplifying Radical Expressions

Many radical expressions contain radicands or factors of radicands that are powers.

When these powers and the index share a common factor, rational exponents can be used to simplify the radical expression.



Simplifying Radical ExpressionsSimplifying Radical Expressions

1. Convert radical expressions to exponential expressions.

2. Use arithmetic and the laws of exponents to simplify.

3. Convert back to radical notation when appropriate.

CAUTIONCAUTION: This procedure works only when all expressions under radicals are nonnegative since rational exponents are not defined otherwise. With this assumption, no absolute-value signs will be needed.



Example Example Radical Exponents Radical Exponents

Use rational exponents to simplify.a. b.8 4x 8 4 6a b

8 4 4/8x x1/ 2xx

SOLUTIONa. b.

1/88 4 6 4 6a b a b

1/ 2 3/ 4a b

4/8 6/8a b

2/ 4 3/ 4a b

1/ 42 3a b

2 34 a b




Use rational exponents to simplify. Do not use exponents that are fractions in the final answer.

SOLUTION

2 2 / 44a) (3 ) (3 )x x1/ 2(3 ) 3x x

Convert to exponential notation

Simplify the exponent andreturn to radical notation




SOLUTION

92 2 9 / 33b) ( )xy z xy z

1/ 41/ 2 1/ 2 1/ 844 8c) y y y y y

2 3 3 6 3( )xy z x y z




Write a single radical expression for

SOLN3/ 4 5/8

1/ 6 1/ 4

x y

x y

3/ 4 1/ 6 5/8 1/ 4x y 9/12 2/12 5/8 2/8x y 7 /12 3/8x y 14/ 24 9/ 24x y

14 924 x y

4161

8543

yx

yx



Rules of Exponents SummaryRules of Exponents Summary

1 , n

n

aa

n na b

b a

1 ,n

n

aa

Assume that no denominators are 0, that a and b are real numbers, and that m and n are integers.

Zero as an exponent: a0 = 1, where a ≠ 0.

00 is indeterminate. Negative exponents:

Product rule for exponents: Quotient rule for exponents: Raising a power to a power: Raising a product to a power: Raising a quotient to a power:

nmnm aaa m n m na a a

nm mna a

n n nab a b

n

n

na ab b



Simplification GuideLinesSimplification GuideLines

The GuideLines for Simplifying expressions with Rational Exponents

1. No parentheses appear

2. No powers are raised to powers

3. Each Base Occurs only Once

4. No negative or zero exponents appear



Example Example Use Exponent Rules Use Exponent Rules Rewrite all radicals as exponentials, and then

apply the rules for rational exponents. Leave answers in exponential form. Assume c > 0

Convert to rational exponents.

Quotient rule

Write exponents with a common

denominator

4 c

c3

= c1/4

c3/2

= c1/4 – 3/2

= c1/4 – 6/4

= c–5/4

=c5/4

1Definition of negative exponent



WhiteBoard WorkWhiteBoard Work

Problems From §7.2 Exercise Set• 58, 74, 78, 106, 110, 112, 132

America’sCup “ClassRule” 5.0Formula



All Done for TodayAll Done for Today

RadicalIndex

Radicand



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

-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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