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Exponents and Radicals

Exponential Notation
If a is any real number and n is a positive integer, then the n^th 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

Radicals
means b² = a and b ≥ 0.

Example.
, since 3² = 9 and 3 ≥0.
Definition of n^th Root
If n > 1 is a positive integer, then the principal n^th 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 n^th 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)