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**Unformatted text preview: **Lesson 2
Exponential Notation:
Where is the base and is the exponent or power Product Rule for Exponents:
When common bases are multiplied, the exponents are added
Quotient Rule for Exponents: When common bases are divided, the exponents are subtracted
(exponent in numerator minus exponent in denominator)
Power Rule for Exponents:
()
When a base is raised to a power and then raised to another
power, the exponents are multiplied
Product to a Power Rule:
(
)
(
)
When a product is raised to a power, the exponent is distributed
to each factor Quotient to a Power Rule:
( ) ( ) When a quotient is raised to a power, the exponent is distributed
to each factor in the numerator and denominator
Zero-Exponent Rule:
(
)
Any base taken to the power of zero is 1, except a base of zero;
Negative Exponent Rule: ( ) (
)
To change the sign of an exponent, take the reciprocal of the
base; if a quantity has a negative exponent, take the reciprocal of
the entire quantity; keep in mind that the sign of the base DOES
NOT change
Radical Notation:
Where is the radicand,
radical sign √
is the index or root and √ is the Definition of √
- the value that must be multiplied by itself
produce
(√ √ )
( (√ √
√ times to ) (√ )
) Think of this as undoing exponential notation
If the index is even, the radicand
nonnegative. Why?
If the index is odd, the radicand
positive, negative or zero. Why? and the radical √ must be
and the radical √ can be Example 1: Evaluate each expression
a. √
b. √
c. √
d. √ e. √ Using the definition, if
, then
in for , we have the following:
√
√
√ || Why?
Example 2: Evaluate each expression
)
a. √( b. (√ c. √( ) ) d. √ e. √( ) √ ; if we substitute In part b. of the previous example, we saw that (√
((√ ) ) ). Given that this is true, what is √ exponential notation? In other words, √
( in . ) Why?
Go back and re-work part b. from the previous example.
(√
Definition of
equivalent to
√ exists ) – the radical expressions √
, where and ( √ ) are is a positive integer greater than and Example 3: Convert the following expressions to radicals and
simplify
a. b. c. d. e. ( ) The laws of exponents are true for rational (fractional)
exponents
Example 4: Rewrite the radicals using rational exponents, then
use exponent rules to separate the radicands
a. √ b. √ Product Rule for Radicals:
- when indices are the same, radicands can be multiplied
(only works if roots exist); when indices are not the same,
use rational exponents;
Simplifying radicals:
- remove factors from the radical until no factor in the
radicand has a degree greater than or equal to the index Example 5: Simplify
a. √ b. √ √ Rationalizing denominators:
- The process by which a fraction is rewritten so that the
denominator contains only rational numbers
Example 6: Rationalized the denominator
a.
√ b. √ c. √ ...

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