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# Roots, Radicals, and Root Functions

• Rational exponents and their relationship to radicals
• Complex numbers

Section 10.1: Radical Expressions and Graphs

• Square Roots

• Cube Roots

• Fourth Roots

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
2.
3.
Also,

Examples: Find each square root.

Examples: Find the square of each radical expression.

Notes:

• 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.

Properties

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.

• 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