Exponents and Radicals
Division of Radicals
Exponents and Radicals
Radicals and Rational Exponents
Radical Equations
Solving Radical Equations
Roots and Radicals
Simplifying Radical Expressions
Radical Expressions
Solving Radical Equations
Solving Radical Equations
Exponents and Radicals
Exponents and Radicals
Roots;Rational Exponents;Radical Equations
Solving and graphing radical equations
Solving Radical Equations
Radicals and Rational Exponents
Roots, Radicals, and Root Functions
Multiplication of Radicals
Solving Radical Equations
Radical Expressions and Equations
Equations Containing Radicals and Complex Numbers
Square Roots and Radicals
Solving Radical Equations in One Variable Algebraically
Polynomials and Radicals
Roots,Radicals,and Fractional Exponents
Adding, Subtracting, and Multiplying Radical Expressions
Square Formula and Powers with Radicals
Simplifying Radicals
Exponents and Radicals Practice
Solving Radical Equations
Solving Radical Equations
Solving Radical Equations
Lecture-Radical Expressions
Radical Functions
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Roots, Radicals, and Root Functions

• Definition of radical expressions
• Graphing radical expressions
• Rational exponents and their relationship to radicals
• Simplifying radical expressions; addition, subtraction, multiplication, division
• Solving radical equations
• Complex numbers

Section 10.1: Radical Expressions and Graphs

• Square Roots

• Cube Roots

• Fourth Roots


Square Roots of a
If a is a positive real number, then
1. the positive or principle square root of a
2. the negative square root of a

For nonnegative a
1. not a real number; it is a complex number

Examples: Find each square root.

Examples: Find the square of each radical expression.


• If a is a positive real number that is not a perfect square, then irrational

• If a is a positive real number that is the square of a number, then rational

• If a is negative real number then, a complex number or a non real number

Examples: Determine whether each square root is rational, irrational or not a real number.

Examples: Find each root.

Graphing Functions Defined by Radical Expressions

• Use a table of values to find points which satisfy the function

• The domain of radical expressions is all values for which the radical is defined

• In radicals with an even index, the radicand must be nonnegative

• In radicals with an odd index, the radicand can be wither positive or negative

Examples: Graph the following radical functions. Give the domain and range in each case.

Simplifying nth Roots

• If n is an even positive integer, then

• If n is an odd positive integer, then

Examples: Simplify each root.

Section 10.2: Rational Exponents

• Relationship between exponents and radicals

• Radicals can be written as rational exponents

Note: If a real number, then

Examples: Evaluate each exponential. Rewrite as a radical.


1. If m and n are integers with in lowest terms, then

2. If is a real number, then

If all indicated roots are real numbers, then

Examples: Simplify each exponential. Rewrite as a radical

Examples: Write each radical as an exponential.

Rules of Exponents

1. Product Rule:

2. Zero Exponent:

3. Negative Exponent:

4. Negative Exponent with Fractions:

5. Quotient Rule:

6. Power Rules:

Examples: Fully simplify the given expressions. Write with only positive exponents. Assume
that all variables represent positive real numbers.

Section 10.3: Simplifying Radical Expressions
• Methods used to simplify radical expressions

Product Rule

• If and are real numbers and n is a natural number, then

• The product of two radicals is the radical of the product

Quotient Rule

• If and are real numbers, with and n is a natural number, then

• The radical of the quotient is the quotient of the radical

NOTE: We can only use the product rule and the quotient rule when the index on the radicals is the same

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