Algebra

Showing posts with label algebra 2. Show all posts
Showing posts with label algebra 2. Show all posts

Free Algebra Study Guide with Videos

Welcome to OpenAlgebra.com, your comprehensive online destination for mastering algebra. Dive deep into our extensive collection of step-by-step tutorials, complemented by illustrative YouTube videos for a visual learning experience. Whether you're a student seeking supplemental material beyond your textbooks or a teacher looking for a rich resource of worked-out math problems, our platform offers a treasure trove of algebraic knowledge tailored to enhance your understanding.
 Chapter 1 - Real Numbers and Their Operations    1.1 Real Numbers and The Number Line    1.2 Adding and Subtracting Integers    1.3 Multiplying and Dividing Integers    1.4 Fractions    1.5 Review of Decimals and Percents    1.6 Exponents and Square Roots    1.7 Order of Operations    1.8 Sample Exam Questions Chapter 2 - Linear Equations and Inequalities    2.1 Introduction to Algebra    2.2 Simplifying Algebraic Expressions    2.3 Linear Equations: Part I    2.4 Linear Equations: Part II    2.5 Applications of Linear Equations    2.6 Ratio and Proportion Applications    2.7 Introduction to Inequalities and Interval Notation    2.8 Linear Inequalities (one variable)    2.9 Review Exercises and Sample Exam Chapter 3 - Graphing Lines    3.1 Rectangular Coordinate System    3.2 Graph by Plotting Points    3.3 Graph using Intercepts    3.4 Graph using the y-intercept and Slope    3.5 Finding Linear Equations    3.6 Parallel and Perpendicular Lines    3.7 Introduction to Functions    3.8 Linear Inequalities (Two Variables)    3.9 Review Exercises and Sample Exam Chapter 4 - Solving Linear Systems    4.1 Solving Linear Systems by Graphing    4.2 Solving Linear Systems by Substitution    4.3 Solving Linear Systems by Elimination    4.4 Applications of Linear Systems    4.5 Solving Systems of Linear Inequalities (Two Variables)    4.6 Review Exercises and Sample Exam    [ Elementary Algebra Exam #1 Solutions ]    [ Elementary Algebra Exam #2 Solutions ]    [ Elementary Algebra Exam #3 Solutions ]    [ Elementary Algebra Exam #4 Solutions ]     --- EA Sample Final Exam Ch. 1-7 --- Chapter 5 - Polynomials and Their Operations    5.1 Rules of Exponents (Integer Exponents)    5.2 Introduction to Polynomials and Evaluating    5.3 Adding and Subtracting Polynomials    5.4 Multiplying Polynomials and Special Products    5.5 Dividing Polynomials  (Synthetic Division)    5.6 Negative Exponents and Scientific Notation    5.7 Review Exercises and Sample Exam Chapter 6 - Factoring and Solving by Factoring    6.1 Introduction to Factoring    6.2 Factoring Trinomials    6.3 Factoring Trinomials ax^2 + bx + c    6.4 Factoring Binomials    6.5 General Guidelines for Factoring Polynomials    6.6 Solving Equations by Factoring    6.7 Applications involving Quadratic Equations    6.8 Review Exercises and Sample Exam Chapter 7 - Rational Expressions and Equations    7.1 Simplifying Rational Expressions    7.2 Multiplying and Dividing Rational Expressions    7.3 Adding and Subtracting Rational Expressions    7.4 Complex Fractions    7.5 Solving Rational Equations    7.6 Applications of Rational Equations    7.7 Variation    7.8 Review Exercises and Sample Exam Do you find this cool math site helpful? If so please share the link or embed in your mylab math course. Also, feel free to copy-and-paste anything you find useful here.  All we ask is that you link back to this site:  OpenAlgebra.com

Embark on an advanced journey with our Intermediate Algebra (IA) section, also known as Algebra 2. Delve into intricate topics, unravel complex equations, and enhance your mathematical prowess. Our meticulously crafted resources, combined with illustrative video tutorials, ensure a deeper understanding of algebraic concepts, bridging the gap between foundational knowledge and advanced applications. Whether you're gearing up for a challenging exam or simply eager to expand your algebraic horizons, our Intermediate Algebra section is tailored to meet your needs.

Solving Exponential Equations

In this section, we will make use of what we have learned about exponential functions to solve equations.

Make use of the above property if you are able to express both sides of the equation in terms of the same base.

Step 1: Express both sides in terms of the same base.
Step 2: Equate the exponents.
Step 3: Solve the resulting equation.

Solve.
It is not always the case that we will be able to express both sides of an equation in terms of the same base.  For this reason we will make use of the following property.
Make use of the above property if you are unable to express both sides of the equation in terms of the same base.

Step 1: Isolate the exponential and then apply the logarithm to both sides.
Step 2: Apply the power rule for logarithms and write the exponent as a factor of the base.
Step 3: Solve the resulting equation.
Solve.

When solving exponential equations and using the above process, the rule of thumb is to choose the common logarithm unless the equation involves the natural base e.   We choose these because there is a button for them on the calculator. However, we could certainly choose any base that we wish; this is the basis for the derivation of the change of base formula.

Properties of the Logarithm

The following properties of the logarithm are derived from the rules of exponents.

The properties that follow below are derived from the fact that the logarithm is defined as the inverse of the corresponding exponential.

To familiarize ourselves with the properties we will first expand the following logarithms. (Assume all variables are positive.)

Expand.
Notice that there is no explicit property that allows us to work with nth roots within the argument of the logarithm.  To simplify these, first change the root to a rational exponent then apply the power rule.
When expanding, notice that we must use the same base throughout the expression. For the next set of problems we will first use the properties to expand then substitute in the appropriate values as the last step.
Evaluate
Expanding is useful for learning the rules and properties associated with logarithms but as it turns out, in practice, condensing down to a single logarithm is the skill that we really need to focus on.

Rewrite as a single logarithm (condense).
Tip: When simplifying these down to one logarithm use only one operation at a time and work from left to right. Combining or skipping steps usually leads to mistakes. Do not race, work slowly, and pay attention to notation.

Evaluate (Round to the nearest ten thousandths where appropriate).
Simplify.
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