Rational Exponents Two - Intermediate Algebra - Lecture Slides, Slides of Algebra

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§7.2 Rational

Exponents

Review §

 Any QUESTIONS About

• §7.2 → Radical Functions

 Any QUESTIONS About HomeWork

• §7.2 → HW-

MTH 55

Example  Laws of Exponents

• Use the rules of exponents to simplify. Write

the answer with

only positive exponents

5/ 6

1/ 6

y

y

−

 SOLUTION

Use 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/ 6 y

=

= y

Example  Laws of Exponents

• Use the Laws of Exponents to Simplify

2 / 5 1/ 5 a. 7 ⋅ 7

1/ 2

1/ 4

b.

m

m

( )

3 / 4 1/ 2 1/ 3 c. x y

−

 SOLUTION

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

b)

m m m m m

− − = = =

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

⋅ = =

Example  Laws of Exponents

• Write with only positive exponents. Assume

that all variables are ≥ 0

Power-to-Power rule

m1/4^ n– m–8^ n2/

–3/

( m–8^ ) –3/4^ (n2/3^ ) –3/

(m1/4^ ) –3/4^ (n–6^ ) –3/

= m^6 n–1/

m–3/16^ n9/

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

= m–99/16^ n^5

Definition of Negative exponent

= m99/

n^5

Product to Power

Example  Laws of Exponents

• Write with only positive exponents. All

variables represent positive numbers x3/5^ (x–1/2^ – x3/4^ ) (^) = x 3/5^ · x–1/2^ – x 3/5^ · x3/4^ Distributive property

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

= x1/10^ – x27/

 Do not make the common mistake of

multiplying exponents in the first step.

Simplifying Radical Expressions

1. Convert radical expressions to exponential

expressions.

1. Use arithmetic and the laws of exponents to

simplify.

1. Convert back to radical notation when

appropriate.

 CAUTION: 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  Radical Exponents

• Use rational exponents to simplify.

a. (^8) x 4 b.^8 a b^4

8 4 4 / 8 x = x 1/ 2 = x

= x

 SOLUTION

a. b.^

( )

8 4 6 4 6 1/ 8 a b = a b

1/ 2 3/ 4 = a b

4 / 8 6 / 8 = a b

2 / 4 3/ 4 = a b

( )

2 3 1/ 4 = a b

4 2 3 = a b

Example  Radical Exponents

• SOLUTION

9 3 2 b) xy z

     

c) 4 y

b) xy z ( xy z)

    =  

( )

1/ 4 4 4 1/ 2^ 1/ 2^ 1/ 8 8 c) y = y = y = y = y

= ( xy z ) = x y z

Example  Radical Exponents

• Write a single

radical expression for

 SOLN

3/ 4 5/ 8

1/ 6 1/ 4

x y

x y

⋅

⋅

3/ 4 1/ 6 5/ 8 1/ 4 x y

− − = ⋅

9 /12 2 /12 5/ 8 2 / 8 x y

− − = ⋅ 7 /12 3/ 8 = x ⋅ y 14 / 24 9 / 24 = x ⋅ y

24 14 9 = x ⋅ y

1 6 1 4

3 4 5 8

x y

x y

⋅

⋅

Simplification GuideLines

• The GuideLines for Simplifying

expressions with Rational Exponents

1. No parentheses appear 2. No powers are raised to powers

1. Each Base Occurs only **Once
2. No negative** or zero exponents appear

Example  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

c^3

= c

1/ c3/

= c1/4 – 3/

= c1/4 – 6/

= c–5/

= (^) c5/

1 Definition of negative exponent

All Done for Today

Radical

Index

Radicand

Bruce Mayer, PE

Licensed Electrical & Mechanical Engineer [email protected]

Chabot Mathematics

Appendix

r − s ≡^ (^ r − s)(r^ + s)

2 2