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Rational Expressions and Their Simplification
Radical Expressions and Equations
Algebraic Expressions
Simplifying Algebraic Expressions
Rational Expressions and Functio
Rational Expressions and Functions
Radical Expressions
Rational Expressions Worksheet
Adding and Subtracting Rational Expressions
Rational Expressions
Multiplying and Dividing Rational Expressions
Dividing Fractions, Mixed Numbers, and Rational Expressions
Multiplying and Dividing Rational Expressions
Multiplying and Dividing Rational Expressions
Simplifying Rational Expressions
Complex Rational Expressions
Rational Expressions and Equations
Integration of Polynomial Rational Expressions
Algebraic Expressions
Radical Expressions & Radical Functions
Rational Class and Expression Evaluator
Adding and Subtracting Rational Expressions
Rational Expressions
Radical Expressions
Multiplying Rational Expressions
Rational Expressions and Common Denominators
rational expressions
Polynomial Expressions
Rational Functions, and Multiplying and Dividing Rational Expressions
Simplifying Radical Expressions
Adding and Subtracting Rational Expressions
Rational Expressions and Equations
Rational Expressions
RATIONAL EXPRESSIONS II
Simplifying Expressions
Quadratic Expressions,Equations and Functions
RATIONAL EXPRESSIONS
Absolute Value and Radical Expressions,Equations and Functions
Rational Expressions & Functions

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Radical Expressions

Square Roots

Square Root
 The number c is a square root of a if c2=a

Ex. Find the roots of…

Principal Square Root
  The principal square root of a nonnegative number is its nonnegative square root.The symbol is called a radical sign and is used to indicate the principal square root of the number over which it appears


Ex. Find the principal root of…

Properties of Square Roots
o Every positive real number has _________ real-numbered square roots
o The square root of zero is _________
o The square root of a negative number is _________ *
o The principal square root of a non-negative number is its _________

Graphs of radical functions and their domains

Ex. Graph f (x) =and find the domain

Ex. Graph g(x) =and find the domain

What is the principal square root of a2
Ex. Find
 

Simplifying
For any real number a ,

(The principal square root of a2 is the absolute value of a.)


Ex. Find the principal square roots

Cube Roots

Cube Root
The number c is the cube root of a if c3 =a.In symbols,we write to denote the cube root of a.

Ex. Find the cube roots of…

Ex. Graph f (x) =and find the domain

Ex. Simplifying the Cube Roots

What are the nth roots?


• The fourth root of a number a is the number c for which c4 = a .There are
also 5th roots, 6th roots, and so on.
• We writefor the nth root.
• The number n is called the index (plural, indices). When the index is 2, we
do not write it.


Summary of Roots

Section 10.2 Rational Exponents

What are rational exponents?


means when a is nonnegative,n can be any natural number greater 1.when a is negative ,n must be odd.

Positive Rational Exponents
For any natural numbers m and n(n≠1) and any real number a for which exists
  and ,or


 

Negative Rational Exponents

For any rational number m/n and any nonzero real number a for which am/n exists
  means


Radicals, Rational Exponents and Calculators

Ex. Approximate the following on your calculator. Round to 4 decimal places

Ex. Re-write each expression as a radical and evaluate

Laws of exponents
For any real numbers a and b and any rational exponents m and n for which am,an and bm are defined
1.am•an = am+n
2.
am/an =am-n
3.(am)n=amn
4.(ab)m=ambm
In multiplying,add exponents if the bases are the same
In deviding,extract exponents if the bases are the same.(Assume a≠0.)
To raise a power to a power ,multiply the exponents
To raise a product to a power,raise each factor to the power and multiply.

Ex. Simplify…

Ex. Use rational exponents to simplify … write as a radical, when appropriate


Section 10.3 Simplifying Radical Expressions

Multiplying and Simplifying Radical Expressions
 

The Product Rule for Radicals
For any real numbers and

(The product of nth roots is the nth root of the product of the two radicands)

Ex. Multiply

Ex. Divide

Simplifying by factoring

To simplify a Radical expression with Index n by Factoring.
1.Express the radicand  as a product in which one factor is the largest perfect nth power possible .
2.Take the nth root of each factor.
3.Simplication is complete when no radicand has a factor that is a perfect nth power.

Ex. Simplify

Ex. Simplify

Ex. Multiply and Simplify

Section 10.4 Dividing Radicals and Rationalizing Denominators

The Quotient Rule for Radicand
For any real numbers and and b≠0

Rationalizing Denominators

Section 10.5 Expressions Containing Several Radical Terms

Adding and Subtracting Radical Expressions
• Adding Radicals is very similar to adding polynomials – "combining like radicals"


Ex. Simplify the radical expressions

More Multiplication and Division
• Use the Distributive Property, FOIL, etc…


Ex. Simplify the radical expressions

Section 10.8 Imaginary and Complex Numbers
 

The number i

i is the unique number for which i=and i2=-1

Ex. Find…

 

Powers of i

Ex. Find…