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 RationalExponents
Bruce Mayer, PE Licensed Electrical Mechanical
EngineerBMayer_at_ChabotCollege.edu
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Review

• Any QUESTIONS About
• 7.2 ? Radical Functions
• Any QUESTIONS About HomeWork
• 7.2 ? HW-32

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Laws of Exponents

• For any real number a, any real number b gt 0,
  and any rational exponents m n.

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.
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Example ? Laws of Exponents

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

• SOLUTION

Use the quotient for exponents. (Subtract the
exponents.)
Rewrite the subtraction as addition.
Add the exponents.
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Example ? Laws of Exponents

• Use the Laws of Exponents to Simplify

• SOLUTION

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Example ? Laws of Exponents

• Use the Laws of Exponents to Simplify

• SOLUTION

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Example ? Laws of Exponents

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

Product to Power
Power rule
Quotient rule
Definition of Negative exponent
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Example ? Laws of Exponents

• Write with only positive exponents. Assume that
  all variables represent

x3/5(x1/2 x3/4)
x3/5 x1/2 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.

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Simplifying 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 expression.

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Simplifying 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.

• 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.

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Example ? Radical Exponents

• Use rational exponents to simplify.
• a. b.

• SOLUTION
• a.

b.
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Example ? Radical Exponents

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

• SOLUTION

Convert to exponential notation
Simplify the exponent andreturn to
radical notation
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Example ? Radical Exponents

• SOLUTION

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Example ? Radical Exponents

• Write a single radical expression for

• SOLN

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Rules of Exponents Summary

• 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

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Simplification GuideLines

• The GuideLines for Simplifying expressions with
  Rational Exponents
• No parentheses appear
• No powers are raised to powers
• Each Base Occurs only Once
• No negative or zero exponents appear

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Example ? Use Exponent Rules

• Rewrite all radicals as exponentials, and then
  apply the rules for rational exponents. Leave
  answers in exponential form. Assume c gt 0

Convert to rational exponents.
Quotient rule
c1/4 3/2
Write exponents with a common denominator
c1/4 6/4
c5/4
Definition of negative exponent
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WhiteBoard Work

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

• AmericasCup ClassRule 5.0Formula

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All Done for Today
RadicalIndexRadicand
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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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