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,
• 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
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
• 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
• The domain of radical expressions is all values for
which the radical is defined
• In radicals with an even index, the radicand must be
• 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
Examples: Evaluate each exponential. Rewrite as a
1. If m and n are integers with
in lowest terms, then
2. If is a real number,
If all indicated roots are real numbers, then
Examples: Simplify each exponential. Rewrite as a
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
• If and
are real numbers and n is a natural number,
• The product of two radicals is the radical of the
• If and
are real numbers, with
and n is a natural number, then
• The radical of the quotient is the quotient of the
NOTE: We can only use the product rule and the
quotient rule when the index on the radicals is the same