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Friday 21st of June
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 Depdendent Variable

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 Dependent Variable

 Number of inequalities to solve: 23456789
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Exponential Notation
If a is any real number and n is a positive integer, then the nth power of a is

The number a is called the base and n is called the exponent.

Zero and Negative Exponents
If a ≠ 0 is any real number and n is a positive integer, then

Laws of Exponents

Example 1. Evaluate the expressions

Example 2. Simplify the expressions

Solution.

 law 4 and 3 rearrange factors factor 6 and use law 1 definition of negative exponents

means b2 = a and b ≥ 0.

Example.
,
since 32 = 9 and 3 ≥0.
Definition of nth Root
If n > 1 is a positive integer, then the principal nth root of a, denoted by is
(1) 0 if a = 0.
(2) the positive number b such that bn = a, if a is positive.
(3) (a) the negative number b such that bn = a, if n is odd and a is negative.
(b) not a real number if n is even and a is positive.

If n is even, we must have a ≥ 0 and b ≥ 0. Complex numbers are needed to define if
a < 0 and n is an even positive integer, because when n is even for all real numbers b, we
have bn ≥ 0.
If n = 2 we write instead of .
Example. since 33 = 81 and 3 ≥ 0.
Example. since (-2)3 = -8.
Note that are not defined as real numbers.

Properties of nth Roots

Note that we have . For example,

Rational Exponents
Definition. If m and n are integers and n > 1, and if a is a real number such that
exists, we define
or equivalently
Note that by the definition
Remark. The laws of exponents hold for rational exponents also.
Example Evaluate the expressions

Example Simplify the expressions

Example Simplify the expression and eliminate any negative exponent(s).

Rationalizing the Denominator
Rationalizing the denominator is to eliminate the radical in a denominator by multiplying
both numerator and denominator by an appropriate expression.
Example.
In general, if the denominator is of the form n with m < n, then multiplying the
numerator and denominator by
Example Rationalize the denominator

More Examples. Simplify the expressions and eliminate any negative exponent(s)