section p.2: exponents and radicals properties of ... p... · 5 example #6: simplifying odd roots...
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Date: ____________________
Section P.2: Exponents and Radicals
Properties of Exponents:
���� � ���� � �� �
���� � ���� � � �� �
Example #1: Simplify.
a.) �3����4���� � b.) 2����� � c.) 3��4���� �
d.) ���� �� �
Example #2: Simplify.
a.) �� � b.) �
���� �
c.) ���� ������ � d.) ���
� �� �
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Square Root: Principal nth Root:
√��
Example #3: Simplify.
a.) √49 =
b.) �√49 �
c.) "���#�
� =
d.) √�32$ =
e.) √�8� �
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Properties of Radicals:
√��� �
" √��� � Example #4: Simplify.
a.) √8 • √2 =
b.) √5� �� �
c.) √��� �
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Simplifying Radicals: An expression involving radicals is in simplest form when the following conditions are satisfied:
1.) All possible factors have been removed from the radical. 2.) All fractions have radical-free denominators (rationalizing the
denominator – not in this class!) 3.) The index of the radical is reduced.
Example #5: Simplifying Even Roots
a.) 484
b.) 75x 3
c.) (5x)44
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Example #6: Simplifying Odd Roots
a.) 243
b.) 24a43
c.) −40x 63
Rational Exponents: Definition of Rational Exponents: If a is a real number and n is a positive integer such that the principal nth root of a exists, we define to be
a1
n
a1
n =
6
If m is a positive integer that has no common factor with n, then
and
Example #7: Changing from Radical to Exponential Form
a.) 3
b.) 3xy( )5
c.) 2x x 34
Example #8: Changing from Exponential to Radical Form
a.) x 2 + y 2( )3
2
am
n = a1
n
m
= amn( ) am
n = am( )1
n = amn
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b.) 2y3
4 z1
4
c.) a−3
2
d.) x 0.2
Example #9: Simplifying with Rational Exponents
a.) 27( )2
6
b.) −32( )−4
5
8
c.) −5x
5
3
3x
−3
4
d.) a39
e.) 1253
f.) 2x −1( )4
3 2x −1( )−1
3
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Example #10: Combining Radicals
a.) 2 48 − 3 27
b.) 16x3 − 54x 43
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Date: ____________________
Section P.3: Polynomials and Factoring
Polynomials: In standard form, a polynomial is written with descending powers of x. The highest exponent in the polynomial is the degree, and the number in front of that term is the leading coefficient. The number in the polynomial without a variable is called the constant term. Example #1: Writing Polynomials in Standard form.
4x2 − 5x7 − 2 + 3x Example #2: Sums and Differences of Polynomials
(7x 4 − x 2 − 4x + 2) − (3x 4 − 4x 2 + 3x)
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Example #3: Multiplying Polynomials – The FOIL Method
(3x − 2)(5x + 7) Example #4: The Product of Two Trinomials
(x + y − 2)(x + y + 2) Example #5: Removing Common Factors
a.) 3x 3 + 9x 2
b.) (x − 2)(2x) + (x − 2)(3)
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Example #6: Removing a Common Factor First
3 −12x2
Example #7: Factoring the Difference of Two Squares
a.) (x + 2)2 − y 2
b.) 16x4 − 81
Example #8: Factoring Perfect Square Trinomials
a.) 16x 2 + 8x +1
b.) x2 −10x + 25
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Example #11: Factoring a Trinomial: Leading Coefficient Is 1
x 2 − 7x +12 Example #12: Factoring a Trinomial: Leading Coefficient Is Not 1
2x 2 + x −15 Example #13: Factoring by Grouping
x3 − 2x2 − 3x + 6
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Date: ____________________
Section P.4: Fractional Expressions Domain of an Algebraic Expression: The set of real numbers for which an algebraic expression is defined is the domain. Example #1: Finding the Domain of an Algebraic Expression
a.) The domain of the polynomial: 2x3 + 3x + 4 is…
b.) The domain of the radical expression x − 2 is…
c.) The domain of the expression
x + 2x − 3 is…
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Example #2: Reducing a Rational Expression
Write
x 2 + 4x −12
3x − 6 in reduced form.
Simplifying Rational Expressions: Example #3: Reducing Rational Expressions
a.)
x 3 − 4 x
x 2 + x − 2
b.)
12 + x − x 2
2x 2 − 9x + 4
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Operations with Rational Expressions: Example #4: Multiplying Rational Expressions
2x 2 + x − 6x 2 + 4 x − 5
•x 3 − 3x 2 + 2x
4 x 2 − 6x
Example #5: Dividing Rational Expressions
x 3 − 8x 2 − 4
÷x 2 + 2x + 4
x 3 + 8
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Example #6: Subtracting Rational Expressions
x
x − 3−
2
3x + 4
Example #7: Combining Rational Expressions: The LCD Method
3
x −1−
2
x+
x + 3
x 2 −1
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Compound Fractions: Example #8: Simplifying a Compound Fraction
2x
− 3
1 −1
x −1
Example #9: Simplifying an Expression with Negative Exponents
x(1 − 2x)−3
2 + (1 − 2x)−1
2
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Example #10: Simplifying a Compound Fraction
(4 − x 2)1
2 + x 2(4 − x 2)−1
2
4 − x 2
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Date: ____________________
Section P.5: Solving Equations
Linear Equations: Example #1: Solving a Linear Equation
Solve 3x − 6 = 0 Example #2: An Equation Involving Fractional Expressions
Solve
x
3+
3x
4= 2
Example #3: An Equation with an Extraneous Solution
Solve
1
x − 2=
3
x + 2−
6x
x 2 − 4
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Example #4: Solving Quadratic Equations by Factoring
a.) 2x 2 + 9x + 7 = 3
b.) 6x 2 − 3x = 0
Example #5: Extracting Square Roots
a.) 4 x 2 = 12
b.) x − 3( )2 = 7
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Example #6: The Quadratic Formula: Two Distinct Solutions
Use the Quadratic Formula to solve: x 2 + 3x = 9
Example #7: The Quadratic Formula: One Repeated Solution
Use the Quadratic Formula to solve: 8x 2 − 24 x +18 = 0
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Polynomial Equations of Higher Degree: Example #8: Solving a Polynomial Equation by Factoring
Solve 3x 4 = 48 x 2
Example #9: Solving a Polynomial Equation by Factoring
Solve x3 − 3x 2 − 3x + 9 = 0
Radical Equations: Example #10: Solving an Equation Involving a Rational Exponent
Solve 4 x 3 2 − 8 = 0
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Example #11: Solving an Equation Involving a Radical
Solve 2x + 7 − x = 2
Absolute Value Equations: Example #12: Solving an Equation Involving Absolute Value
Solve x 2 − 3x = −4 x + 6