# data[].attributes.body
1. Step 1: Put the equation in Slope Intercept Form (y=mx+b).

2. Step 2: Graph the y-intercept point (the number in the b position) on the y-axis. ...

3. Step 3: From the point plotted on the y-axis, use the slope to find your second point. ...

4. Step 4: Draw your line using the two points you plotted (y-intercept (b) first, slope (m) second.

# data[].attributes.headline
Another example for the Explicit Equation:

# data[].attributes.subject
Another example for the Explicit Equation:

# data[].attributes.headline
How to write the Explicit Equation for a Geometric Sequence:

# data[].attributes.subject
How to write the Explicit Equation for a Geometric Sequence:

# data[].attributes.headline
How to write the Explicit Equation for an Arithmetic Sequence:

# data[].attributes.subject
How to write the Explicit Equation for an Arithmetic Sequence:

# data[].attributes.headline
Answers to Practice finding the Sequence Generator (multiplier):

# data[].attributes.subject
Answers to Practice finding the Sequence Generator (multiplier):

# data[].attributes.headline
Answers to the Practice finding the Sequence Generator:

# data[].attributes.subject
Answers to the Practice finding the Sequence Generator:

# data[].attributes.headline
Common Ratio: This is the multiplier! The number that is multiplied by each term to get the next term.  

# data[].attributes.subject
Common Ratio: This is the multiplier! The number that is multiplied by each term to get the next term.  

# data[].attributes.headline
Practice Finding the Sequence Generator (multiplier):

# data[].attributes.subject
Practice Finding the Sequence Generator (multiplier):

# data[].attributes.headline
Common Difference: The number that is being added or subtracted each time in order to get the next term.

# data[].attributes.subject
Common Difference: The number that is being added or subtracted each time in order to get the next term.

# data[].attributes.headline
Practice finding the Sequence Generator:

# data[].attributes.subject
Practice finding the Sequence Generator:

# data[].attributes.headline
VIDEO: How to figure out the "formula" for arithmetic sequences:

# data[].attributes.subject
VIDEO: How to figure out the "formula" for arithmetic sequences:

# data[].attributes.body
Definition

# data[].attributes.headline
Writing the Equation with 2 Points (another way

# data[].attributes.subject
Writing the Equation with 2 Points (another way

# data[].attributes.attachment_link.title
writing exponential equations given two points

# data[].attributes.attachment_link.description
writing exponential equations given two points, exponential equations, algebra 2

# data[].attributes.attachment_link.description
NERDSTUDY.COM for more detailed lessons! Let's learn about Asymptotes.

# data[].attributes.headline
Writing an Exponential Equation from a Table

# data[].attributes.subject
Writing an Exponential Equation from a Table

# data[].attributes.attachment_link.title
How to Write an Exponential Function from a Table

# data[].attributes.attachment_link.description
Uploaded by Ms. Jaros Algebra I on 2016-03-20.

# data[].attributes.headline
Writing an Exponential Equation from a Graph

# data[].attributes.subject
Writing an Exponential Equation from a Graph

# data[].attributes.attachment_link.title
Writing Exponential Functions from a Graph

# data[].attributes.attachment_link.description
Practice Writing the Equation of an Exponential Function Given it's Graph

# data[].attributes.headline
Writing an Exponential Equation from a Situation

# data[].attributes.subject
Writing an Exponential Equation from a Situation

# data[].attributes.attachment_link.description
Uploaded by cwilson maths on 2016-11-18.

# data[].attributes.attachment_link.readable_text
The graph below is an example of a step function. As you examine the graph, determine why you think it might be called a step function.

Do you see what looks like a set of steps? This is one reason why it is called a step function. It is better known as a discontinuous function.

Why do you think it is called a discontinuous function? Yes, it is not a continuous line, it stops and starts repeatedly.

So, the question may be, is it a function? Does it pass the vertical line test? Let's see!

It looks like the vertical lines may touch two points on the graph at the same time. However, take a look at the points. One is a closed circle and one is an open circle.

If you review our inequalities lesson, you will remember that a closed circle means that the point includes that particular point. But... an open circle does NOT include that point.

So, in this case, where it looks like the vertical line is touching two points, it is really only touching one point, because the open circle does not include that point. So, to answer our question, yes this is considered a function. It's not linear, and it's not quadratic. We call it a step function or a discontinuous function.

How Do We Read and Interpret a Discontinuous Graph?

Let's take a look at our postage graph again.

This graph describes how much it will cost to send a letter depending on the weight of the letter. I've labeled the steps so that you better understand the explanation below.

Step 1: If the weight of the letter is over 0 oz and up to 1 oz (including 1 oz, since the circle is closed), it will cost 39 cents.

Step 2: If the weight of the letter is more than 1 oz (not 1 oz exactly because the circle is open) and up to 2 oz (including 2 oz since the circle is closed), then the price is 41 cents.

Step 3: If the weight of the letter is more than 2 oz (not 2 oz exactly because the circle is open) and up to 3 oz (including 3 oz since the circle is closed), then the price is 43 cents.

Steps 4-6 follow the same pattern as steps 1-3 described above.

As you can see, this graph tells you exactly how much your letter will cost depending on the weight. A discontinuous graph must be used because the price stays the same between ounces, but then changes to the next price as you reach a whole ounce.

Let's take a look at a few other discontinuous graphs and determine whether or not they are functions. These graphs may not look like "steps", but they are considered discontinuous.

This graph is not a function because when utilizing the vertical line test, it touches in two points. Both points at x = 1 are solid, therefore the graph is discontinuous, but not a function.

This graph is a function because it passes the vertical line test. Each vertical line only touches the graph at one point. (Although it looks like it touches at two points at x = -3, since one circle is "open" we do not include that as a point.) Therefore, it is considered a discontinuous function. It is discontinuous at x = -3.

Let's practice creating and interpreting a graph for a discontinuous function.

Example 1
A wholesale t-shirt manufacturer charges the following prices for t-shirt orders:

$20 per shirt for shirt orders up to 20 shirts.

$15 per shirt for shirt between 21 and 40 shirts.

$10 per shirt for shirt orders between 41 and 80 shirts.

$5 per shirt for shirt orders over 80 shirts.

Sketch a graph of this discontinuous function.

You've ordered 40 shirts and must pay shipping fees of $10. How much is your total order?

Solution

If I ordered 40 shirts and must pay $10 in shipping fees, then my total order will cost $610. (40 * $15) +10 = 610.

Example 2

This would be a great problem for you to try on your own:

In many states a "sales tax" is added to most goods that you buy. The tax rate varies from state to state. Let's suppose that your particular state issues a sales tax on any goods purchased.

You are selling candy bars. The taxable amounts and tax imposed up to $1 are shown below.

For amounts between $0.01 and $0.20, the tax is $.01.

For amounts greater than $0.20 and less than or equal to $0.40, the tax is $0.02.

For amounts greater than $0.40 and less than or equal to $0.60, the tax is $0.03.

For amounts greater than $0.60 and less than or equal to $0.80, the tax is $0.04

For amounts greater than $0.80 and less than or equal to $1.00, the tax is $0.05.

Complete the graph to show the tax that imposed on candy bars.

Use the graph to answer the following questions:

1. A candy bar costs $0.55. What is the total cost with tax?

2. Your aunt purchased three candy bars at $0.55 a piece. What is the total cost with tax?

3. Someone purchased 4 candy bars at $0.55 a piece. They gave you $2 and a quarter. Is this enough money to cover the candy bars and the tax? Explain your answer.

Answer Key

The following is the completed graph for the discontinuous funtion.

The red circles indicate open circles.

1. If a candy bar costs $0.55, then the total cost with tax is $0.58. (0.55 +0.03)

2. The total cost of three candy bars is $1.74.

3. If someone gave me $2.25 for 4 candy bars, they would not have given me enough money. The total cost would be $2.31

Great Job! You have successfully completed the Step functions lesson.

You Might Also Like These Function Lessons:
Top of the Page

# data[].attributes.attachment_link.title
Step Functions & Discontinuous Functions

# data[].attributes.attachment_link.description
The graph below is an example of a step function. As you examine the graph, determine why you think it might be called a step function. Do you see what looks like a set of steps? This is one reason why it is called a step function.

# data[].attributes.attachment_link.description
You can look at different types of graphs in terms of how much of the data they actually show. The dot plot shows all of it: every value appears individually. Our age histogram groups the data into 19 classes, so it distills 5000 numbers down to only 19.

# data[].attributes.headline
Constructing Histograms, Dot Plots, and Box Plots

# data[].attributes.subject
Constructing Histograms, Dot Plots, and Box Plots

# data[].attributes.attachment_link.description
This project was created with Explain Everything ™ Interactive Whiteboard for iPad.

# data[].attributes.attachment_link.description
Read and learn for free about the following article: Mean, median, and mode review

# data[].attributes.attachment_link.readable_text
Essential Question: How do I summarize, represent, and interpret data on a single count or measurement variable?

Lesson

Data can be represented using a variety of graph types. Graphs provide a visual representation of data; the spread and center can be determined quickly by looking at the graph. Dot plots and histograms show the frequency of a data value, and box and whisker plots show only the five-number summary of the data (smallest value, lower quartile, median, upper quartile, and largest value).

In a dot plot, each data value is represented by a dot or other graphic. The number of times a value is repeated corresponds to the number of dots above that value. Dot plots make it easy to see gaps and clusters in a data set, as well as how the data spread along the axis.

Histograms usually gather data into categories, allowing a quick comparison for each category. In a histogram, the heights of a rectangle above a value correspond to the number of data values with that value. If a data set is large, a histogram is easier to use because a single dot does not need to be drawn for each data value. There is no set number cutoff for when a data set is too large for a dot plot, but some sources that for more than 20 - 30 data items a histogram is preferred.

For instance, Bob is on the basketball team and scored the following number of points per game for the last 40 games: 8, 15, 10, 10, 10, 15, 7, 8, 10, 9, 12, 11, 11, 13, 7, 8, 9, 9, 8, 10, 11, 14, 11, 10, 9, 12, 14, 14, 12, 13, 5, 13, 9, 11, 12, 13, 10, 8, 7, 8.

The data can be shown in any of these graph types:

Once the data are graphed, analyze the shape using the following terms:

Center: Graphically, the center of distribution is the point(s) where about half the observations are on either side.

Spread: The spread of a distribution refers to the variability of the data. If the observations cover a wide range, the spread is larger. If the observations are clustered around a single value, the spread is smaller. Data tend to cluster about the mean of a data distribution.

Shape: The shape of a distribution is described by symmetry, number of peaks, direction of skew, or uniformity. Real-life data may vary slightly from these perfect examples, look for overall shapes and trends in the graphs. Symmetry- A symmetric distribution can be divided at the center so that each half is a mirror image of the other.

Number of Peaks: Distributions can have few or many peaks. Distributions with one clear peak are called unimodal and distributions with two clear peaks are called bimodal. Unimodal distributions are sometimes called bell-shaped.

Direction of Skew: Some distributions have many more observations on one side of graph than the other. Distributions with a tail on the right toward the higher values are said to be skewed right; and distributions with a tail on the left toward the lower values are said to be skewed left.

Uniformity: When observations in a set of data are equally spread across the range of the distribution; the distribution is called uniform distribution. A uniform distribution has no clear peaks.

Unusual Features: Unusual features refer to gaps (areas of the distribution where there are no observations) and outliers. Outliers are data points that lie far away from the main cluster of points.

Examples

Example 1. John works in a pencil factory. He randomly selects 100 pencils per day and measures them to the nearest millimeter. Explain which type of graph John should choose to represent his data.

Solution: There are many correct answers to a problem like this, so the justification should thoroughly back up the choice. Since the number of pencils is large, John may want to consider a histogram so that 100 dots would not need to be drawn. Since it did not say exactly what information John was hoping to get from his graph, the box-and-whisker point may not be the best choice because it gives only a summary of the information.

Example 2. An accounting firm looked at the exam scores of the Certified Public Accountants test of its applicants and recorded the results in the distribution below. Describe the distribution by its symmetry, number of peaks, direction of skew and uniformity.

Solution: Real-life data often has slight variations, look at the overall trends and shape of the graph. The shape of the graph is relatively symmetric with a line at 40 - 50 dividing the graph into two mirror images. It has one clear peak 40 - 50, and is not skewed either way.

The graph is symmetric, unimodal, and bell shaped.

Example 3. Anna and Ethan survey 20 students to determine how many colds that they have had this year. The results are summarized in the table.

Part a: Which type of graph would best model the situation? Solution: There is not one clear answer for this problem, as long as the answer is justified completely. This example has used a histogram because a separate dot for each data point is not necessary.

Part b: Construct the graph for the data. Solution: Start by making a grid that goes from 0 - 7+ on the x-axis. Draw rectangles to the height correspond to the number of students.

Part c: Describe the graph by its shape, number of peaks, and direction of skew. Solution: Its shape is non-symmetric, the fact that it has only one peak makes it unimodal, and since most of the data is on the left the graph is skewed right.

Example 4:. Each girl in Mr. Sanson's class and in Mrs. Kwei's class measured her own height. The heights were plotted on the dot plots below.

Part a: Compare the centers of the data sets and determine which class likely has a larger mean height.

Solution: The center of the data distribution is the point where about half the observations are on either side, which corresponds to the median. For Mr. Sanson's class, it is appears visually that about half the observations lie on each side of 67. Confirm by determining the median, there are 25 students, so the median will occur at the 13 th data point. For Mrs. Kwei's class, it is appears visually that about half the observations lie on each side of 65. Confirm by determining the median, there are 25 students, so the median will occur at the 13 th data point.

The heights in Mr. Sanson's class appear to cluster around 67 inches, and those in Mrs. Kwei's class cluster around 65 inches, so Mr. Sanson's class likely has a larger mean height.

Part b: Compare the spread of the classes.

Solution: The spread of a distribution refers to the variability of the data. If the observations cover a wide range, the spread is larger. If the observations are clustered around a single value, the spread is smaller. The students in Mrs. Kwei's class seem to be clustered more closely around the middle values.

Self-Check

Self Check Problem 1. Dr. Jones wants to analyze the pulse rate of her adult patients. She wants to understand the center and spread of her data so that she has a better understanding of an expected pulse rate for an adult. Which graph would be most useful to Dr. Jones? Solution

Self Check Problem 2. Anna and Ethan watch 10 thirty-minute shows during the month of June. They record the number of food commercials that air during each show in the table below. Create a dot plot for the data. Solution

Self Check Problem 3. The distribution below shows the heights of 30 people. Describe the graph by its symmetry, number of peaks, direction of skew, or uniformity. Solution

Self Check Problem 4. Both ABC Company and XYZ Company offer its employees early retirement options (retirement before age 65). There were 19 employees at each company who accepted this option last year. Distributions are shown below.

Part a: Estimate the center of the distribution for the ABC Company. Solution

Part b: Estimate the center of the distribution for the ZYX Company. Solution

Part c: Which company likely has a greater average retirement age? Solution

# data[].attributes.attachment_link.description
Essential Question: How do I summarize, represent, and interpret data on a single count or measurement variable? Data can be represented using a variety of graph types. Graphs provide a visual representation of data; the spread and center can be determined quickly by looking at the graph.

# data[].attributes.headline
Two Way Frequency Tables and Relative Frequency

# data[].attributes.subject
Two Way Frequency Tables and Relative Frequency

# data[].attributes.attachment_link.readable_text
You have worked with one-way tables (even though you may not have called them by that name). A one-way table is simply the data from a bar graph put into table form. In a one-way table, you are only working with one categorical variable .

You can probably guess that a two-way frequency table will deal with two variables (referred to as bivariate data). In so doing, these tables examine the relationships between the two categorical variables. Two-way frequency tables are especially important because they are often used to analyze survey results. Two-way frequency tables are also called contingency tables .

The Basics of a Two-Way Frequency Table

Two-way frequency tables are a visual representation of the possible relationships between two sets of categorical data. The categories are labeled at the top and the left side of the table, with the frequency (count) information appearing in the four (or more) interior cells of the table. The "totals" of each row appear at the right, and the "totals" of each column appear at the bottom. "sum of the row totals"
equals the "sum of the column totals" Note: the (the 240 seen in the lower right corner). This value (240) is also the sum of all of the counts from the interior cells.

A survey asked, "If you could have a new vehicle, would you want a sport utility vehicle or a sports car?

Let's take a look at the vocabulary used to identify cell locations in two-way frequency tables. Entries in the body of the table (the blue cells where the initial counts appear) are called joint frequencies. The cells which contain the sum (the orange "Totals" cells) of the initial counts by row and by column are called marginal frequencies. Note that the lower right corner cell (the total of all the counts) is not labeled as a marginal frequency.

When a two-way table displays percentages or ratios (called relative frequencies), instead of just frequency counts, the table is referred to as a two-way These two-way tables can show relative frequencies for the whole table, for rows, or for columns. Notice that the relative frequencies may be displayed as a ratio, a decimal ( relative frequency table. to nearest hundredth), or percent ( to nearest percent).

Relative Frequency for Whole Table:

If the two-way relative frequency is for the whole table, each entry in the table is divided by the total count (found in the lower right corner). The ratio of "1", or 100%, occurs only in the cell in the lower right corner.
Each of the main body cells (blue) is telling you the percentage of people surveyed that gave that response (based upon the total number of people responding).

So what's up with more women, than men, choosing a sports car?

It is certainly possible that women may love sporty cars just as much, or more, than men. While this may be possible, it is not the real situation regarding this survey data. The misleading information is that the frequency table and the relative frequency table shown above do not take into consideration how many women, and how many men, responded to this survey. There were only 60 men responding, while there were 180 women. There were three times more women responding to this survey, which presents misleading results when based upon the entire population.

To avoid such problems when comparing the categorical variables in a two-way frequency table, we need to exam the table by separate categories (rows or columns). When a relative frequency is determined based upon a row or column, it is called a "conditional" relative frequency. To obtain a conditional relative frequency, divide a joint frequency (count inside the table) by a marginal frequency total (outer edge) that represents the condition being investigated. You may also see this term stated as row conditional relative frequency or column conditional relative frequency.
Basically, we are going to look at the women and men separately, based upon how many women were surveyed, and how many men were surveyed.

Conditional Relative Frequency for Rows:

If we want to answer the question, "In this survey, do more men, or more women, prefer a sports car?", we need to set up a row conditional relative frequency. The listings of men and women are row headings, and we want to examine these categories separately to determine the answer to our question. By choosing a row method, we are comparing men and women in relation to car type.

Conditional Relative Frequency for Columns:

What we have seen, by examining all of these tables, is that different tables answer different types of questions about the data. If you want to look for a relationship between the categorical variables, you will need to prepare a conditional relative frequency table. You will then need to decide if a "row" method or a "column" method will address the situation you wish to examine.

A variety of questions can be answered by examining a two-way frequency table.

Let's look at some possibilities:

Two-way frequency table

What is the relative frequency of males choosing a sports car?

Associations Based on Conditional Relative Frequency:

An "association" exists between two categorical variables if the row (or column) conditional relative frequencies are different for the rows (or columns) of the table. The bigger the differences in the conditional relative frequencies, the stronger the association between the variables. If the conditional relative frequencies are nearly equal for all categories, there may be no association between the variables. Such variables are said to be independent.

In our Sports Car and SUV example (above), the row conditonal relative frequencies showed a good degree of difference. Based upon that information, if we knew the gender of a survey respondent, we could make a good prediction as to whether he/she chose a sports car or an SUV. The statistical information is strong enough to support an "association" between gender and choice of vehicle. Now, this does not mean that there is always an association between gender and choice of vehicle. It just means that such an association is evident in the data from this survey.

# data[].attributes.attachment_link.title
Two-Way Frequency Tables - MathBitsNotebook(A1 - CCSS Math)

# data[].attributes.attachment_link.description
You have worked with one-way tables (even though you may not have called them by that name). A one-way table is simply the data from a bar graph put into table form. In a one-way table, you are only working with one categorical variable .

# data[].attributes.attachment_link.title
Standard Deviation - Explained and Visualized

# data[].attributes.attachment_link.description
What are residuals and what do they mean?

# data[].attributes.attachment_link.title
Graphing Piecewise Functions, Domain & Range - Limits, Continuity, & Absolute Value ,

# data[].attributes.attachment_link.readable_text
To limit the domain or range (x or y values of a graph), you can add the restriction to the end of your equation in curly brackets {}. For example, y=2x{1<x<3} would graph the line y=2x for x values between 1 and 3.

You can also use restrictions on the range of a function and any defined parameter.

It's also possible to add multiple restrictions to the same expression line regardless of what parameter is being restricted.

Have more questions? Submit a request

# data[].attributes.attachment_link.description
To limit the domain or range (x or y values of a graph), you can add the restriction to the end of your equation in curly brackets {}. For example, y=2x{1<x<3} would graph the line y=2x for x values between 1 and 3.

# data[].attributes.attachment_link.title
How to Solve Single Variable Inequalities , Intermediate Algebra , Lesson 46

# data[].attributes.attachment_link.description
This tutorial explains how to solve a single variable inequality. Need More Algebra Help? I highly recommend this book. http://amzn.to/1EApcPp Donate http://bit.ly/19AHMvX

# data[].attributes.attachment_link.title
How To Graph Linear Inequalities In Two Variables - Basic Introduction, Algebra

# data[].attributes.attachment_link.title
Introduction to graphing systems of linear inequalities | Algebra II | Khan Academy

# data[].attributes.headline
Word Problems to Understand the Solutions to Inequalities

# data[].attributes.subject
Word Problems to Understand the Solutions to Inequalities

# data[].attributes.attachment_link.title
Systems of linear inequalities word problems example | Algebra II | Khan Academy

# data[].attributes.headline
Solving Quadratics with the Factoring Method

# data[].attributes.subject
Solving Quadratics with the Factoring Method

# data[].attributes.attachment_link.entities[].name
Factoring Binomials Using Difference of Perfect Squares Method

# data[].attributes.attachment_link.title
Solving Quadratic Equations By Factoring - Basic Examples, Quadratic Formula - Algebra

# data[].attributes.attachment_link.description
This algebra video tutorial shows you how to solve quadratic equations by factoring. It shows you how to factor quadratic expressions using the difference of squares method, the ac method, factoring by grouping, removing the gcf - greatest common factor, factoring polynomials with 3 terms or factoring trinomials.

# data[].attributes.headline
Solving Quadratics with the Square Root Method

# data[].attributes.subject
Solving Quadratics with the Square Root Method

# data[].attributes.attachment_link.title
Solving Quadratic Equations with the Square Root Method, part 1 5-6a

# data[].attributes.headline
Solving Quadratics with the Quadratic Formula Method

# data[].attributes.subject
Solving Quadratics with the Quadratic Formula Method

# data[].attributes.attachment_link.title
Solve Quadratic Equations using Quadratic Formula

# data[].attributes.headline
Solving Quadratics with the Completing the Square Method

# data[].attributes.subject
Solving Quadratics with the Completing the Square Method

# data[].attributes.attachment_link.title
Completing The Square Method and Solving Quadratic Equations - Algebra 2

# data[].attributes.attachment_link.title
Ex:  Find the Equation of a Quadratic Function from a Graph

# data[].attributes.attachment_link.description
This video explains how to determine the equation of a quadratic function from a graph. It used the standard form of a quadratic function and then write the standard form in general form. Complete Library: http://mathispower4u.com Search by Topic: http://mathispower4u.wordpress.com

# data[].attributes.attachment_link.readable_text
Rules of Exponents: There are 4 basic exponent rules. Students in grade 7 and beyond need to be familiar with these rules and should be able to apply these rules when dealing with problems involving exponents.
Image Source
Zero Exponent Rule or Power Zero Rule: This is a very very basic rule to remember as most often overlooked by students and which cost them lot of marks in their math quizzes and exams. The rule is that "If any number raised to the power of zero then its value is equal to one".

For example; consider the problem, 2º which is read as two raised to the power zero. The answer is equal to 1 as shown below:

2º = 1

Below are more example on zero exponent law

5º = 1

9º = 1

Or in case of algebra (5xy)º = 1

Make sure you do it right when there involves negative number as the base. Look at the example below to understand how the negative sign can change the answer when parenthesis are used and when not used:

2. Law of Exponent Multiplication: In case there are two exponents being multiplied and their bases are the same then we can add the powers (exponents) to get the new power (exponent) and base remains the same. For example;
5² x 5 {we can write it as follows as 5 can be written as 5¹} = 5² x 5¹ { add the powers 2 and 1 to get the new power as 3 and base remains the same}
= 5³

3. Law of Exponent Division: If we divide two exponential terms and their bases are same then we subtract the power of the denominator from the power of the numerator to get the new power and base remains the same. For example;
3³/3² = 3¹ {But 3¹ is same as 3 hence we can write the final answer as shown below}
= 3

4. Exponent of an exponent: There is another rule involving exponents which is exponent of another exponents. These problems look like (3³)² and to solve it just multiply both the exponents to get the new exponent and base remains the same. We can solve it as follows

(3³)² = 3³˙² = 3^6 (which is 3 raised to the power of 6)

Below is a practice sheet to practice all the laws of exponents
Click for pdf format of above worksheet

# data[].attributes.attachment_link.description
Rules of Exponents: There are 4 basic exponent rules. Students in grade 7 and beyond need to be familiar with these rules and should be able to apply these rules when dealing with problems involving exponents.

# data[].attributes.attachment_link.description
The Math Dude obeys The Laws of Exponents as he multiplies monomials, takes a power of a power, and raises a monomial to a power.

# data[].attributes.headline
Simplifying Expressions: Distributing and Combining Like Terms

# data[].attributes.subject
Simplifying Expressions: Distributing and Combining Like Terms

# data[].attributes.headline
Function Web: Equation, Table, Graph, Situation

# data[].attributes.subject
Function Web: Equation, Table, Graph, Situation

# data[].attributes.attachment_link.title
Literal Equations - MathHelp.com - Algebra Help

# data[].attributes.body
Multiply Everything!

# data[].attributes.attachment_link.title
How to Graph Equations with Slope Intercept Form

# data[].attributes.attachment_link.title
Solving a System of Equations Using Elimination and Multipliers

# data[].attributes.attachment_link.title
Learn to solve a system of equations using substitution

# data[].attributes.attachment_link.title
Algebra - Solving Linear Equations by using the Graphing Method 1/2

# data[].attributes.attachment_link.readable_text
Factors

Numbers have factors:

And expressions (like x²+4x+3) also have factors:

Factoring

Factoring (called " Factorising" in the UK) is the process of finding the factors:

Factoring: Finding what to multiply together to get an expression.

It is like "splitting" an expression into a multiplication of simpler expressions.

Example: factor 2y+6

Both 2y and 6 have a common factor of 2:

So we can factor the whole expression into:

2y+6 = 2(y+3)

So 2y+6 has been "factored into" 2 and y+3

Factoring is also the opposite of Expanding:

Common Factor

In the previous example we saw that 2y and 6 had a common factor of 2

But to do the job properly we need the highest common factor, including any variables

Firstly, 3 and 12 have a common factor of 3.

So we could have:

But we can do better!

Together that makes 3y:

So we can factor the whole expression into:

3y ²+12y = 3y(y+4)

Check: 3y(y+4) = 3y × y + 3y × 4 = 3y²+12y

More Complicated Factoring

Factoring Can Be Hard !

The examples have been simple so far, but factoring can be very tricky.

Because we have to figure what got multiplied to produce the expression we are given!

It is like trying to find which ingredients
went into a cake to make it so delicious.
It can be hard to figure out!

Experience Helps

With more experience factoring becomes easier.

Example: Factor 4x² − 9

Hmmm... there don't seem to be any common factors.

But knowing the Special Binomial Products gives us a clue called the "difference of squares":

So we have:

And that can be produced by the difference of squares formula:

So let us try doing that:

Yes!

Answer: 4x ² − 9 = (2x+3)(2x−3)

How can you learn to do that? By getting lots of practice, and knowing "Identities"!

Remember these Identities

Here is a list of common "Identities" (including the "difference of squares" used above).

It is worth remembering these, as they can make factoring easier.

=

(a+b)(a−b)

=

(a+b) (a+b)

=

(a−b) (a−b)

=

=

a ³+3a ²b+3ab ²+b ³

=

(a+b) ³

a ³−3a ²b+3ab ²−b ³

=

(a−b) ³

There are many more like those, but those are the most useful ones.

Advice

The factored form is usually best.

When trying to factor, follow these steps:

There are also Computer Algebra Systems (called "CAS") such as Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, Reduce and others that can do factoring.

More Examples

Experience does help, so here are more examples to help you on the way:

Example: w⁴ − 16

An exponent of 4? Maybe we could try an exponent of 2:

Yes, it is the difference of squares

And "(w ² − 4)" is another difference of squares

That is as far as I can go (unless I use imaginary numbers)

Remove common factor "3u":

Then a difference of cubes:

That is as far as I can go.

Try factoring the first two and second two separately:

z ²(z−1) − 9(z−1)

Wow, (z-1) is on both, so let us use that:

And is a difference of squares

(z−3)(z+3)(z−1)

That is as far as I can go.

Now get some more experience:

338, 339, 2047, 2048, 2049, 178, 2050, 3177, 3178, 3179

# data[].attributes.attachment_link.description
Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents.

# data[].attributes.attachment_link.title
Writing equations in all forms | Algebra I | Khan Academy

# data[].attributes.attachment_link.title
Interpreting linear graphs word problems example 1 | Algebra I | Khan Academy