W A T K I N S - J O H N S O N C O M P A N Y Semiconductor Equipment Group

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Chabot Mathematics
7.2 RadicalFunctions
Bruce Mayer, PE Licensed Electrical Mechanical
EngineerBMayer_at_ChabotCollege.edu
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Review

• Any QUESTIONS About
• 7.1 ? Cube nth Roots
• Any QUESTIONS About HomeWork
• 7.1 ? HW-30

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Cube 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

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Example ? Cube Root of No.s

• Find Cube Roots
• a) b) c)

• SOLUTION
• a)

As 0.20.20.2 0.008

• b)

As (-13)(-13)(-13) -2197
As 33 27 and 43 64,so (3/4)3 27/64

• c)

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Rational 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
• Recall ? SomeThingSomeThing SomeThing2
• This suggests that a1/2 is a square root of a.

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Definition of a1/n

• 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.

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nth 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 omitted.

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Evaluating a1/n

• Evaluate Each Expression

(a)
271/3
3

(b)
641/2
8

(c)
6251/4
5

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

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Caveat on Roots

• CAUTION 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
5

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

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Radical Functions

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

• Example ? Given f(x) Then find
  f(3).

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

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Example ? Exponent to Radical

• Write an equivalent expression using RADICAL
  notation
• a) b) c)

• SOLUTION

a)
c)
b)
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Example ? Radical to Exponent

• Write an equivalent expression using EXPONENT
  notation
• a) b)

• SOLUTION

a)
b)
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Exponent ? IndexBase ? Radicand

• From the Previous Examples Notice

The denominator of the exponent becomes the
index. The base becomes the radicand.
The index becomes the denominator of the
exponent. The radicand becomes the base.
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Definition of am/n

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

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Example ? am/n Radicals

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

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Example ? am/n Exponents

• Rewrite with rational exponents

• SOLUTION

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Definition of a-m/n

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

• That is, am/n and a-m/n are reciprocals

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Caveat on Negative Exponents

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

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Example ? Negative Exponents

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

• SOLUTION

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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

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Example ? 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?
• SOLUTIONSubstitute 74 for t in the Formula and
  find an approximation using a calculator.

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WhiteBoard Work

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

• The MACH No.M

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All Done for Today
Ernst MachFluidDynamicist

• Born 8Feb1838 in Brno, Austria

• Died 19Feb1916 (aged 78) in Munich, Germany

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Chabot Mathematics
Appendix
Bruce Mayer, PE Licensed Electrical Mechanical
EngineerBMayer_at_ChabotCollege.edu

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Graph y x

• Make T-table

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