fundamental concepts of algebra unit 1. exponents and radicals lesson 1.2
TRANSCRIPT
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FUNDAMENTAL CONCEPTS OF ALGEBRA
Unit 1
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EXPONENTS AND RADICALS
Lesson 1.2
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Lesson Essential Question (LEQ)
How do we simplify radicals and algebraic expressions with rational exponents?
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Laws of Exponents for Real Numbers
A)
B)
C)
D)
E)
**(Blue Tables on Page 20)**(Always simplify negative exponents completely!)
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Fractional Exponents
or
**(Green Table on Page 27)**
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Simplifying Expressions with Exponents
Examples:
Textbook Page 29 #’s 16, 22, 32, 38, 44, 46
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Homework:
Page 29 #’s 11-45 odds only
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Bell Work:
Simplify:
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Radicals
The whole thing is called a radical.
The n represents the index or root of the radical.
The a represents the radicand.
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You need to know:
Squares:
Cubes:
Prime Factorization:
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Laws of Radicals
Page 24 in your textbook.
These are properties and laws of radicals that you should file into your long-term memory!!!
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Removing/Simplifying nth Powers
How would we simplify the following?
A)
B)
C)
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Rationalizing Denominators
It is impossible to divide by an irrational number (which most radicals are), so we have to rationalize the denominator of any fractional expression that has a radical in it. **(Blue Table on Page 26)**
Ex:
Ex: :
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Homework:
Pages 29-30 #’s 53-79 odds only
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Bell Work:
Simplify each expression completely:
1)
2)
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Class Examples:
Textbook Pages 29-30 #’s 24, 30, 44, 64, 70, 74, 76, 78, 80, 100
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Homework:
Review Blue Tables on Pages 32, 36, and 38.
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Bell Work:
Simplify the expression:
2)
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ALGEBRAIC EXPRESSIONS
Lesson 1.3
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Lesson Essential Question (LEQ)
How do we perform operations and factor algebraic expressions?
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Polynomials
Adding
Subtracting
Multiplying
Dividing
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Polynomial Product Formulas
**(Blue Table on Page 36)**
Hint: If you know these, it will save you time!
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Polynomial Examples
Textbook Page 43 #’s 4, 10, 14, 18, 28, 34
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Homework:
Pages 43-44 #’s 1-21 odds, 25, 29, 33, 35, 37, 39
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Bell Work:
Simplify:
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Factoring Polynomials
To factor any polynomial, you must first take out the Greatest Common Factor (GCF).
Factor:
When you factor something, you are breaking it down to its original factors.
Examples Page 44 #’s 46 and 50
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Trinomials
In the form , when a = 1. (simple)Ex:
In the form , when a≠ 1. (Master Product Method)
Ex:
Examples Page 44 #’s 54 and 58
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Special Case Binomials:
Difference of Two Squares
Difference of Two Cubes
Sum of Two Cubes
**(Blue Table on Page 38)**
Examples Page 44 #’s 72, 76, 78, 80
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Factoring By Grouping
When you have four different terms that are separated by addition or subtraction, you can group certain terms together in order to factor.
Examples Page 44 #’s 86, 88
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Remember:
It is possible that a polynomial cannot be factored, so in this instance, you would say the polynomial expression is irreducible.
Ex: , this is irreducible.
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Homework:
Page 44 #’s 47, 51, 55, 57, 61, 69, 75, 81, 85, 89, 95
What you need to know for the quiz tomorrow:Powers and RadicalsAdd/Sub/Mult/Div PolynomialsFactoring
Look through the examples we did in class and the problems from previous homework assignments and you should be fine!
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Bonus for the Quiz:
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Bell Work:
Factor:
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FRACTIONAL EXPRESSIONS
Lesson 1.4
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Lesson Essential Question (LEQ)
How do we perform operations with fractional expressions?
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Things to remember:
A fractional expression is a quotient of two algebraic expressions.
Whenever we have a variable in the denominator, we are going to have excluded values. Why?
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Examples of Multiplying/Dividing
Page 54 #’s 8, 12, 14
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Adding and Subtracting Fractional Expressions
MUST HAVE A COMMON DENOMINATOR!!!
If it does not have a common denominator, find it!
Examples: Page 54 #’s 20, 22, 26
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Homework:
Page 54 #’s 5 – 27 odds only
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Bell Work:
Simplify:
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Complex Fractions:
A complex fraction is when you have a quotient in which the numerator and/or the denominator is a fractional expression.
We need to simplify the numerator and denominator as much as possible before we divide them!
Examples: Page 54 #’s 34, 40, 44
Remember that division is the same as multiplying by the reciprocal!!!!!!!!!!!
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Rationalizing
Sometimes it is necessary to rationalize a denominator or numerator of a fractional expression.
We multiply the numerator and denominator by the conjugate.
Examples: Page 55 #’s 52, 56, 60
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Homework:
Pages 54-55 #’s 33, 39, 43, 49, 51, 55
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Bell Work:
Simplify:
1)
2)
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Class Work:
Pages 54 – 55 #’s 14, 20, 26, 28, 30, 36, 42, 44, 50, 76, 77, 78
Work on this in class today and tonight for extra practice with fractional expressions.
We are going to review tomorrow, and have a small test on Monday!!!
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Bell Work:
Simplify:1)
2)
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Unit 1 Test
You need to know:
Exponents/Radicals (1.2)Add/Sub/Mult/Div Polynomials (1.3)Factoring (1.3)Fractional Expressions (1.4)
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Review Problems
We have a Unit 1 test tomorrow on Lessons 1.2-1.4.
Pages 56-57, you should be able to do 13-84.
Today, work on the following in groups:
16, 20, 28, 30, 38, 40, 42, 66, 74, 76, 82, 84
For extra practice, you can try doing the odds at home to prepare!
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Bonus for the Test!
A 5 inch wooden cube is painted blue. The cube is then cut into smaller 1 inch cube pieces. How many of the smaller 1 inch cubes have paint on 3 sides? 2 sides? 1 side? 0 sides?
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