Radical Functions: Cube Roots, Rational Exponents, and Nth Roots, Slides of Algebra

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§7.2 Radical

Functions

Review §

 Any QUESTIONS About

• §7.1 → Cube & nth^ Roots

 Any QUESTIONS About HomeWork

• §7.1 → HW-

7.1 MTH 55

Example  Cube Root of No.s

• Find Cube Roots (^3) a) (^) 0.008 b) c) 3 27 64

 SOLUTION

• a) (^3) 0.008 = 0.2 As 0.2·0.2·0.2 = 0.
• b) (^3) − 2197 = − 13 As (−13)(−13)(−13) = −
• c) 3

27 3 64 4

= As 3so (3/4)^3 = 27 and 4 3 = 27/64^3 = 64,

Rational Exponents

• Consider a 1/2 a 1/2. If we still want to

add exponents when multiplying, it

must follow from the Exponent

PRODUCT RULE that

a 1/2 a 1/2^ = a 1/2 + 1/2, or a^1

» Recall  [SomeThing]·[SomeThing] = [SomeThing]^2

• This suggests that a 1/2^ is a

square root of a.

n th^ Roots

• n th^ root: The number c is an n th^ root of a number a if c n^ = a.
• The fourth root of a number a is the number c for which c^4 = a. We write for the n th^ root. The number n is called the index (plural, indices ). When the index is 2 (for a Square Root), the Index is omitted.

n (^) a

Evaluating a 1/ n

• Evaluate Each Expression (a) 27 1/3^3 27 = 3

(b) 64 1/2^ 64 = 8

=

=

(c) –625 1/4^ = –^4 625 = –

(d) (–625) 1/4^4 –625 is not a real number because the radicand, –625, is negative and the index is even.

=

Radical Functions

• Given PolyNomial, P , a RADICAL FUNCTION Takes this form:  Example  Given f ( x ) = Then find f (3).

( ) n f x = P

5 x −8,

 SOLUTION

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

f (^) ( ) 3 = 5 3( ) − (^8) = 15 − (^8) = 7

Example  Exponent to Radical

• Write an equivalent expression using RADICAL notation

a) b) c)

3

1

m

( ) 2 4 4 4

1 2

1

9 x^8 = 9 x = 9 x = 3 x

( )^5 5

1

xy^2^ z = xy z

 SOLUTION

3 3

1

a) m = m

b)

c)

( )^2

1 9 x^8 ( )^5

1

xy^2 z

Exponent ↔ Index

Base ↔ Radicand

• From the Previous Examples Notice:

3 3

1 m = m

The denominator of the exponent becomes the index. The base becomes the radicand.

( ) 3 1 3 4 x = 4 x

The index becomes the denominator of the exponent. The radicand becomes the base.

Definition of a m / n

 For any natural numbers m and n

( n not 1) and any real number a for

which the radical exists,

( )

/ means , or.

m n (^) n^ m^ n m a a a

n (^) a

Example  a m / n^ Exponents

• Rewrite with rational exponents 3 a) x^5 ( ) 5 2 b) 3 xy

 SOLUTION

( ) (^ ) 5 2 2 / 5 b) 3 xy = 3 xy

(^3 5) 5 / 3 a) x = x

Definition of a−m / n

 For any rational number m/n and

any positive real number a the

NEGATIVE rational exponent:

/ /

1 means.

m n m n

a a

−

 That is, a m / n^ and a − m / n^ are

reciprocals

Example  Negative Exponents

• Rewrite with positive exponents, & simplify
  • a. 8−2/3^ b. 9−3/2 x 1/5^ c. 3 3/ 4 2

t r

−      SOLUTION 3 / 2 1/ 5 1/ 5 3 / 2 b) 9 1 9

− (^) x = ⋅ x

(^1) 1/ 5^ 1/ 5 27 27

= ⋅ x =^ x

3 3 / 4^2 3 / 4 c) 2 3

t r r t

 ^ −     =     

2 / 3 2 / 3 a) 8 1 8

− (^) =

2

1 1 2 4

= =

( ) 3 2

1 8

=

Example  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

S = 21.9 5 t +2457.