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Square Roots and Radical Expressions
Solving Radical Equations
Simplifying Radical Expressions
Irrational Numbers in General and Square Roots in Particular
Roots of Polynomials
Simplifying Radical Expressions
Exponents and Radicals
Products and Quotients Involving Radicals
Roots of Quadratic Equations
Radical Expressions
Radicals and Rational Exponents
Find Square Roots and Compare Real Numbers
Radicals
Radicals and Rational Exponents
Theorems on the Roots of Polynomial Equations
SYNTHETIC DIVISION AND BOUNDS ON ROOTS
Simplifying Radical Expressions
Exponents and Radicals
Properties of Exponents and Square Roots
Solving Radical Equations
Rational Exponents and Radicals,Rationalizing Denominators
Rational Exponents and Radicals,Rationalizing Denominators
Quadratic Roots
Exponents and Roots
Multiplying Radical Expressions
Exponents and Radicals
Solving Radical Equations
Solving Quadratic Equations by Factoring and Extracting Roots
Newton's Method for Finding Roots
Roots of Quadratic Equations Studio
Roots, Radicals, and Root Functions
Review division factoring and Root Finding
Radicals
Simplifying Radical Expressions
Multiplying and Simplifying Radical Expressions
LIKE RADICALS
Multiplication and Division of Radicals
Radical Equations
BOUNDING ROOTS OF POLYNOMIALS
   
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Exponents and Radicals

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


Radicals
  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)

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