Find the indicated root, or state that the expression is
not a real number.
Teaching Notes: • The symbol is called the radical sign .
• The number under the radical sign is called the radicand.
• Together we refer to the radical sign and its radicand as a radical.
• The symbol - is used to denote the negative
square root of a number.
• The square root of a negative number is not a real number. This also applies
to any even
• Not all radicals are square roots.
c. 3.162 d. not a real number 4. a. 3 b. -1 c. not a real number d. -2 e. 2 f.
-4 g. -1
Mini Lecture 9.2 Multiplying and Dividing Radicals
3. Simplify. (Look for a pattern) Assume all variables
represent positive number only.
5. Multiply. Then simplify if possible.
Teaching Notes: • Have students memorize perfect square numbers through 225 and perfect
cubes through 216.
• Get as much out of the radicand as possible.
• Since radicals are unfamiliar to most students, it is important they see the
squaring numbers and square roots, cubing numbers and cube roots, etc.
Mini Lecture 9.3 Operations with Radicals
Learning Objectives: 1. Add and subtract radicals.
2. Multiply radical expressions with more than one term.
3. Multiply conjugates.
Examples: Add or subtract as indicated.
Teaching Notes: • Two or more square roots can be combined using the distributive property
have the same radicand.
• In some cases, radicals can be combined after they have been simplified.
• When multiplying radical expressions, distribute. This is similar to
monomial by a polynomial.
• When multiplying radical expressions use the FOIL method like multiplying
• When multiplying conjugates (expressions that involve the sum and difference
same two terms ), the FOIL method may be used or the special product formula.
using the FOIL method with conjugates the OI (outside & inside) will equal 0.
Answers: e. cannot be combined
Mini Lecture 9.4 Rationalizing the Denominator
Learning Objectives: 1. Rationalize denominators containing one term.
2. Rationalize denominators containing two terms.
Examples: Multiply and simplify.
Rationalize each denominator.
State the conjugate of each of the following.
Rationalize each denominator and write in simplest form.
Teaching Notes: • Remind students of the definition of a rational number. This will help
the meaning of “rationalizing the denominator.”
• It may be helpful to discuss the special product of (a + b)(a - b) with
several examples to
let students “see” again what happens to the middle term when the binomials are
Answers: 1. a. x = 2 b. x = 5 c. x = 3 d. no solution e. -3 f. no solution g. 10
h. no solution
i. 81 j. -2, -1
Mini Lecture 9.6 Rational Exponents
Learning Objectives: 1. Evaluate expressions with rational exponents.
2. Solve problems using models with rational exponents.
Write each of the following in radical form first, then simplify.
Teaching Notes: • If a graphing calculator is being used in the class, it is helpful
to show that is the same
as the using number values.
• Stress to students that the denominator of a rational exponent is the index of
corresponding radical expression.
• When the numerator of a rational exponent is not 1, the numerator is the power
the radical is raised. It is usually easier to simplify it this way, but it is
possible to raise
the radicand to the power instead.
• When the exponent is negative , write the base as its reciprocal, and raise to
After successful completion of MAT 1010, a student should
be able to:
Text: Beginning and Intermediate Algebra, 2nd Edition, by
Course Description: Solution of linear and absolute value equations and
inequalities; integer and rational exponents, simplification of radicals; slope
and graphing linear equations ; systems of linear equations; solution of quadratic
by factoring, completing the square and using the quadratic formula ;
functions; applications included throughout the course. Writing assignments, as
appropriate to the discipline, are part of the course.
Other materials: Students are required to have a scientific or a graphing
You are not allowed to use a cell phone as a calculator.
Internet Resources: The textbook we are using is bundled with MathZone
software . MathZone is a completeonline tutorial and course management system for
mathematics and statistics, designed for greater ease of use than any other
available. MathZone is a powerful Web-based tutorial for homework, quizzing,
and multimedia instruction. Also available in CD-Rom format, MathZone offers:
• Practice Exercises based on the text and generated in unlimited
as much practice as needed to master any objective.
3. [Jun 16-Jun 20]: Least Common Denominator(7.3), Addition and Subtraction of
Rational Expressions(7.4), Complex Fractions(7.5), Rational Equations(7.6),
Applications of Rational Equations and Proportions(7.7).
4. [Jun 23-Jun 27]: Introductions to Relations(8.1), Introductions to
Graphs of Functions(8.3), Compound Inequalities(10.1). [Review & Exam I].
5. [Jun 30-Jul 4]: Absolute Value Equations(10.3), Linear Inequalities in Two
Variables (10.5), Definition of the nth Root(11.1), Rational Exponents(11.2).
6. [Jul 7-Jul 11]: Simplifying Radical Expressions(11.3), Addition and
Radicals(11.4), Multiplications of Radicals (11.5), Rationalization(11.6).
7. [Jul 14-Jul 18]: Radical Equations(11.7), Complex Numbers(11.8), Square Root
Property and Completing the Square(12.1), Quadratic Formula (12.2). [Review & Exam II].
8. [Jul 21-Jul 25]: Equations in Quadratic Form(12.3), Graphs of Quadratic
Functions(12.4), Vertex of Parabola and Applications(12.5). [Review & Final