Download Math 002 – Intermediate Algebra

Math 002 – Intermediate Algebra
Student Notes & Assignments
Unit 4 – Rational Exponents, Radicals, Complex Numbers and Equation Solving
Unit 5
Homework
Topic
Due Date
7.1
BOOK
pg. 491: 62, 64, 66, 72, 78, 80
Radicals & Radical Functions
7.2
BOOK
pg. 498: 12, 14, 30, 48, 50, 56
Rational Exponents
MML
A16 (7.1, 7.2)
7.3
BOOK
pg. 507: 8, 24, 40, 52, 66, 78,
Simplifying Radical Expressions
7.4
BOOK
pg. 512: 10, 30, 52, 60, 64, 66
Add/Subtract/Multiply Radicals
MML
A17 (7.3, 7.4)
BOOK
pg. 520: 10, 12, 26, 36, 46
MML
A18 (7.5)
M: Apr. 22
MML
Quiz 4
M: Apr. 22
7.6
BOOK
pg. 528: 10, 14, 18, 38, 56, 62
Radical Eqns & Problem Solving
7.7
BOOK
pg. 538: 12, 14, 28, 36, 54, 56
Complex Numbers
MML
A19 (7.6, 7.7)
8.1
BOOK
pg. 559: 12, 22, 30, 50, 62,
Completing the Square
8.2
BOOK
pg. 570: 6, 12, 26, 44, 58
The Quadratic Formula
MML
A20 (8.1, 8.2)
7.5
M: Apr. 15
TH: Apr. 18
Rationalize Denominators
TH: Apr 25
M: Apr. 29
EXAM 4 Group A : TH-NOV 29 & Group B : FR – NOV 30
Monday
8-Apr
Tuesday
9-Apr
Wednesday
10-Apr
Week 11
§ 7.1, 7.2
15-Apr
16-Apr
17-Apr
MML #16 (7.1, 7.2)
§ 7.3, 7.4
§ 7.4
22-Apr
23-Apr
MML #18 (7.5)
§ 7.2, 7.3
Week 14
24-Apr
MML Q4
§ 7.7, 8.1
§ 7.6
29 -Apr
MML #20 (8.1, 8.2)
§ 8.2
§ 7.1, 7.2
18-Apr
19-Apr
1-May
Quiz 7 (7.1-7.4)
Quiz 7 (7.1-7.4)
§ 7.5, 7.6
§ 7.5
25-Apr
MML #19 (7.6, 7.7)
Review
7-May
Final Review
26-Apr
Quiz 8 (7.5-7.7)
Quiz 8 (7.5-7.7)
§ 8.1, 8.2
§ 8.1
2-May
Exam 4
Week 15
Final Review
§ 7.7
30-Apr
Review
6-May
Friday
12-Apr
MML #17 (7.3, 7.4)
Week 12
Week 13
Thursday
11-Apr
Final Review
8-May
9-May
RETAKE for Exam 3 or Exam 4
Final Review
Final Review
3-May
Exam 4
Final Review
10-May
STOP DAY
1
Sections 7.1 Radicals and Radical Functions
Objectives:
 Find square roots, cube roots, nth roots.
 Find
where a is a real number.
 Look at the graphs of square root and cube root functions and evaluate function values.
1. Finding Square Roots: EX 1
 Radical expression:

To take a square root means to undo a square.

If a is a nonnegative number, then
o
o
is the principal, or nonnegative, square root of a.
is the negative square root of a.
Example 1: Simplify. Assume that all variables represent positive numbers.
a)
b)
c)
d)
e)
f)
2. Finding
Roots. EX 3, EX 4
Radical expression:

To take an
root means to undo an nth power.
2
Example 2: Simplify each radical. Assume that all variables represent positive real numbers. The index is
important.
#59
#64
#60
#65
#71
Note:
is a rational number.
is an irrational number.
3. The square root and cube root functions. EX 6, EX 7, EX 8
 Domain.
 General Graph:
Basic square root function
Basic cube root function
Example 3: Match the graph with its equation. Also, give the domain.
A.
B.
Example 4: If
a.
C.
and
b.
D.
, find each function value.
c.
d.
3
Section 7.2 - Rational Exponents
Objectives:
 Understand the meaning of
,
and
.
 Use rules for exponents to simplify expressions that contain rational exponents.
 Use rational exponents to simplify radical expressions.
Sometimes we will see radicals expressed as rational exponents.
1. Understand the meaning of
. EX 1
 If a is a real number and n is a positive integer, then
 The quantity
.
is the nth root of a.
 Notice that the denominator of the rational exponent corresponds to the _______ of the radical.
Example 1: Use radical notation to write each expression. Simplify if possible.
a.
b.
c.
d.
e.
f.
2. Understand the meaning of
. EX 2
By properties of exponents,
 We can generalize this: If m and n are positive integers greater than 1 (with
in lowest terms), then
Example 2: Use radical notation to write each expression. Simplify if possible.
a.
b.
c.
d.
4
3. Understand the meaning of
. EX 3
Ask students,
What is
What is
where m and n are integers?
?
What is
?
What is
when n is an integer?

Example 3: Write each expression with a positive exponent, and then simplify.
a.
b.
c.
d.
4. Using rules for exponents to simplify expressions. EX 4, EX 5
The properties of exponents which we reviewed earlier, apply to rational exponents as well.
Let a and b be real numbers
Let m and n be integers
Let m and n be rational numbers
1.
2.
3.
4.
Example 4: Use the properties of exponents to simplify each expression. Write with positive exponents.
a.
b.
c.
d.
e.
#60.
5
Example 5: Use rational exponents to simplify. Assume that variables represent positive numbers.
#76.
#80.
#83.
6
Section 7.3 Simplifying Radical Expressions
Objectives:
 Use the product rule for radicals.
 Use the quotient rule for radicals.
 Simplify radicals.
 Use the distance formula.
1. Using the Product Rule. EX 1
Product Rule for Radicals
If two rational expressions have the same _______________ then when multiplying radicals we can simply
multiply the values under the radicals.
Example 1:
a.
b.
c.
d.
2. Using the Quotient Rule. EX 2
Quotient Rule for Radicals
Example 2:
a.
b.
c.
d.
3. Simplifying Radicals EX 3, EX 4, EX 5
 Both the product and quotient rules can be used to simplify a radical.
Example 3: Simplify. Assume variables represent non-negative values.
a.
b.
c.
d.
e.
f.
7
g.
h.
A radicand in the form
is simplified when the radicand a contains no factors that are perfect nth powers
(other than 1 or -1).
 All factors of the radicand have exponents less than the index.
 The greatest common factor of the index and exponents of all the radicand factors is 1.
4. The distance formula. EX 6
Now that we know how to simplify radicals, we can derive and use the distance formula.
The distance
between two points
and
is given by:
Example 4: Find the distance between each pair of points. Give an exact distance and a three-decimal
approximation.
a.
#75.
8
Section 7.4 – Adding, Subtracting, and Multiplying Radical Expressions
Objectives:
 Add/subtract radical expressions.
 Multiply radical expressions.
1. Like radicals: radicals with the same ________________ and the same _______________.
Complete the table with “like” or “unlike”.
Terms
Like or unlike terms?
1.
2.
3.
4.
5.
2.
Add/Subtract Radical Expressions. EX 1, EX 2, EX 3

Simplify each radical term first

Combine like terms (like radicals).
Example 2: Add or subtract as indicated.
a.
b.
c.
d.
e.
f.
9
#15
3.
#40.
Multiplying Radical Expressions. EX 4
As with multiplying polynomials, many of the same properties are used to multiply expressions with
radical terms.
Example 3: Multiply.
a.
b.
c.
#56.
d.
e.
10
Section 7.5 – Rationalizing Denominators of Radical Expressions
Objectives:
 Rationalize denominators with one term.
 Rationalize denominators having two terms.
Rationalizing the Denominator means rewriting the rational expression without any _______________
in the denominators.
1. Rationalizing the denominator when there is one term in the denominator. EX 1, EX 2, EX 3
 Multiply the numerator and denominator by the expression that will make the denominator the
nth root of a perfect nth power.
 Simplify.
Example 1: Rationalize the denominator.
a.
b.
c.
d.
e.
f.
Quick Review:
These are called conjugates. Because the product results in the difference of two squares, we will need the
conjugates to rationalize the denominator when there are two terms in the denominators.
Example 2: Find the conjugate of each expression, then find their product.
expression
conjugate
product
1.
2.
3.
11
2. Rationalizing Denominators Having Two Terms. EX 4
 Multiply the numerator and denominator by the conjugate of the denominator.
 Simplify.
Example 3: Rationalize each denominator.
a.
b.
c.
#48.
12
Section 7.6 Radical Equations and Problem Solving
Objectives:
 Solve equations that contain radical expressions.
 Use the Pythagorean theorem to model problems.
1. POWER RULE
EX 1, EX 4, EX 5


If both sides of an equation are raised to the same power, all solutions of the original
equation are among solutions of the new equation.

This property does not state that raising both sides of an equation to a power yields an
equivalent equation. A solution of the new equation may or may not be a solution of
the original equation. You must check each solution of the new equation to make sure it
is a true solution or extraneous solution of the original equation.
Example 1: Solve. Show checking both numerically and graphically.
a)
Check:
b)
c)
d)
e)
13
#22.
2. Pythagorean Theorem. EX 6
If and are the lengths of the legs of a right triangle and is the length of the hypotenuse, then
_________________________.
#53. Figure is of a right triangle with side length 3m and hypotenuse 7m – find the length of the other leg.
14
Section 7.7 Complex Numbers
Objectives:
 Write square roots of negative numbers in the formbi.
 Add, subtract, multiply, and divide complex numbers.
 Raise I to powers.
1. Write Square Roots of Negative numbers in the form

. EX 1
The imaginary unit, written as _____, is the whole number whose square is -1.
_____ and
_____ .
Example 1: Write in terms of
a)

Encourage students to write the radical last when simplifying.
c.
b)
Multiplying or dividing with a negative number under a square root. EX 2
Example 2: Multiply or divide. Emphasize that the following is not true:
d)
e)
g)
h)
.
f)
2. Complex Numbers.

A complex number is a number that can be written in the form
real numbers and is the imaginary unit.
, where
are
Example 3: Write the following in complex form.
a.
b.
c.
d.
15

Addition and subtraction of complex numbers
 Add/Subtract like terms
Example 4: Perform the indicated operations.
a.

Products of complex numbers
 Multiply as polynomials, rewrite to basic terms and then combine like terms.
c)

b.
d)
e)
Quotients of complex numbers
 Multiply numerator and denominator by the conjugate of the denominator; rewrite in complex
form; simplify
f)
g)
h)
POWERS of
16
Section 8.1 Solving Quadratic Equations by Completing the Square
Objectives:
 Use the square root property to solve quadratic equations.
 Use quadratic equations to solve problems.
We will continue to look at different ways to solve quadratic equations.
Previously, we learned how to solve a quadratic equation by factoring.
In the next two sections we will look at different ways to solve quadratic equations
1. The Square Root Property. EX 1, EX 2, EX 3, EX 4
If
is a real number and if
, then
_______.
Even when the right side of the equation is not a perfect square, we can use this property.
Example 1: Solve for x.
a.
b.
c.
d.
Review:
Perfect Square Trinomials
Factored Form
2. Completing the Square. EX 5, EX 6, EX 7
This method is such a useful tool to solve quadratic equations, we can change equations that are not
perfect squares on the left-hand sides to perfect squares.
Example 2: What number is needed to make a perfect square trinomial?
a.
____
b.
____
c.
____
d.
____
17
Example 3: Solve by completing the square.
Leading coefficient is 1.
i.
j.
Leading coefficient is not 1.
k.
l.
Example 4: Use the graph to determine how many real number solution(s) exists for each equation.
#72.
#74.
18
Section 8.2 Solving Quadratic Equations by the Quadratic Formula
Objectives:
 Solve quadratic equations by using the quadratic formula.
 Determine the number and type of solutions of a quadratic equation by using the discriminant.
 Solve geometric problems modeled by quadratic equations.
Any quadratic equation can be solved by completing the square. When we complete the square of the general
quadratic equation in standard form
we obtain the Quadratic Formula.
1. Quadratic Formula. EX 1, EX 2

A quadratic equation written in the form
has the solutions
Example 1: Solve.
a.
b.
c.
What are the different possibilities of solutions for a quadratic equation?
19
2. The Discriminant. EX 5

The discriminant is the portion of the quadratic formula under the radical:
what the solutions are going to look like before we solve them.
. It can tell us
Number and Type of solutions
Positive
Zero
Negative
Example 2: Use the discriminant to determine the number and types of solutions of each equation.
You can check with the graphing calculator.
d.
e.
f.
3.
Applications.
#52. Given the diagram below, approximate to the
nearest foot how many feet of walking
distance a person saves by cutting across the
lawn instead of walking on the sidewalk.
54. The hypotenuse of an isosceles right triangle is
one meter longer than either of its legs. Find the
length of each side.
20