Title: W A T K I N S - J O H N S O N C O M P A N Y Semiconductor Equipment Group
1Chabot Mathematics
7.2 RationalExponents
Bruce Mayer, PE Licensed Electrical Mechanical
EngineerBMayer_at_ChabotCollege.edu
2Review
- Any QUESTIONS About
- 7.2 ? Radical Functions
- Any QUESTIONS About HomeWork
- 7.2 ? HW-31
3Laws 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.
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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.
4Example ? Laws of Exponents
- Use the rules of exponents to simplify. Write the
answer with only positive exponents
Use the quotient for exponents. (Subtract the
exponents.)
Rewrite the subtraction as addition.
Add the exponents.
5Example ? Laws of Exponents
- Use the Laws of Exponents to Simplify
6Example ? Laws of Exponents
- Use the Laws of Exponents to Simplify
7Example ? Laws of Exponents
- Write with only positive exponents. Assume that
all variables are 0
Product to Power
Power-to-Power rule
Quotient rule
Definition of Negative exponent
8Example ? Laws of Exponents
- Write with only positive exponents. All variables
represent positive numbers
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.
9Simplifying 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.
10Simplifying Radical Expressions
- Convert radical expressions to exponential
expressions. - Use arithmetic and the laws of exponents to
simplify. - 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.
11Example ? Radical Exponents
- Use rational exponents to simplify.
- a. b.
b.
12Example ? Radical Exponents
- Use rational exponents to simplify. Do not use
exponents that are fractions in the final answer.
Convert to exponential notation
Simplify the exponent andreturn to
radical notation
13Example ? Radical Exponents
14Example ? Radical Exponents
- Write a single radical expression for
15Rules 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
16Simplification 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
17Example ? 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
18WhiteBoard Work
- Problems From 7.2 Exercise Set
- 58, 74, 78, 106, 110, 112, 132
- AmericasCup ClassRule 5.0Formula
19All Done for Today
RadicalIndexRadicand
20Chabot Mathematics
Appendix
Bruce Mayer, PE Licensed Electrical Mechanical
EngineerBMayer_at_ChabotCollege.edu
21Graph y x
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