Exponents - what do they mean?

Exponents are a short cut to express certain types of multiplication. In algebra we sometimes want to multiply a number by itself again, and again, and again. If we wanted to write 8x8, we are multiplying 8 twice.  We would express this as the number 8 raised to the 2nd power.

The number 8 is called the base and the exponent is called the power.






Here are other examples:

When we see a number with an exponent, the exponent is telling us how many times to write the base.  We can use exponents on bases that are numbers or letters (variables).





Some practice problems to make sure you understand the concept:

Write the following using exponent form:

1. X*X*X*X*X

2. 3*3*3*a*a

3. 6*6*X*X*Y*Y*Y

Write the following exponents out to show the expanded form (indicating where multiplication is occurring):

4. 3m⁴n<sup>6

5. 22x4y3

6. 23mn3p2 

Solutions:</sup>

1. X⁵

       2. 3³a²     

       3. 6²X²Y³

4. 3*m*m*m*m*n*n*n*n*n*n
5. 2*2*x*x*x*x*y*y*y
6. 2*2*m*n*n*n*p*p

Here is a link to a short, but useful video explaining how exponents work (click here) - check it out if you need a little more explanation.

Collecting Like Terms

Just because a problem asks you to add (or subtract) terms, doesn't mean we can...

When simplifying or solving algebraic expressions we have to know when we can collect terms and when we can't.

We can only collect (or add/subtract) LIKE terms.  So what are LIKE terms???

Terms are considered to be like terms if they have the same variable (or letter component) and the same exponent.  Let's look at the lists below and see how some terms are like terms and some terms are not.

If we look at the column of Like terms, we can see that the leading number (also known as the coefficient) can be anything, but the letters (variables) and the exponent (or power on the variable) must be identical. Like terms can be added or subtracted from one another (I like to think of them as being in the same family)!

If we look at the column of unlike terms, we can see that either the variables are different or the exponents are different. Unlike terms can not be added or subtracted from one another!!

 For example - when we see a problem like this, it looks like they want us to add and subtract all of these terms and come up with one term as the answer... but we can't!:

 3xy - 4xy + 2xyz

There are two terms that are alike and one that is not alike.  The positive 3xy and the negative 4xy have identical variables and exponents (remember that if no exponent is visible, then the variable is raised to an invisible 1).  The term 2xyz has that extra "z" in its variable, and so it is not identical to the other two terms.

When we collect like terms, we let the numbers and signs do the work, and we bring the variables along.  A positive 3xy and a negative 4xy have different signs, so we will subtract the numbers and keep the sign of the bigger number (the 4 is bigger than the 3, so the negative sign wins).  The difference between 3 and 4 is 1.  So 3xy -4xy = -1xy.

The problem can be simplified to -1xy + 2xyz.  We can't add these last two terms together because they are not LIKE terms.

There is a great video about collecting like terms at Khan Academy (click here).  When the link takes you to Khan Academy there are actually several good videos entitled "collecting like terms", you should watch them all!!  Then there are practice problems you can try (click here).

Knowing when and how to collect like terms is going to very important to succeeding in Algebra!

Orders of Operations

The order of operations is a BIG DEAL.

So maybe you've heard the saying "Please excuse my dear aunt Sally" and thought your math teacher had lost his/her mind.  But that little saying contains a lot of useful information!  The letter at the beginning of each word serves as a reminder about which operations we need to do first in our math problems.

The P in "please" stands for parentheses - this tells us that any work INSIDE a parentheses (or brackets for that matter) must be done first when starting a problem.  If there is no work that can be done inside a set of parentheses, then we move on to the next task. (Be careful not to confuse the parentheses that we sometimes use for multiplication as the parentheses/brackets that this rule is referring to).

The E in "excuse" stands for exponents (radicals, square roots for example are included in this rule).  Any number raised to an exponent should be calculated before you can move on to the rest of the orders of operations.

The M in "my" and the D in "dear" stand for multiplication and division. These two operations are equally important and should be done as you see them from left to right.

The A in "aunt" and the S in "Sally" stand for addition and subtraction.  These two operations are equally important and should be done as you see them from left to right.

As with most things in math, the only way to master a topic is to practice, practice practice.

Here is a website with several videos (click here) and endless practice problems (click here).
Here is a website with written directions and lots of practice problems for you to try (click here).
Here is a website that has a few worked examples for you to study and then some problems for you to try on your own (click here).  At the bottom of the page there is a link that says "Need More Practice - try our order of operations worksheet generator".  You can follow that link and make worksheets for yourself - the website will grade your answers!

Evaluating expressions

Algebra can consist of any combinations of numbers, variables, operators and exponents.  If we know how to interpret the expression we can "evaluate" it.

Evaluating an expression means to find its value (what it's equal to) if we know what each variable represents.

Let's just consider the expression:

3x + 7

This expression says to multiply 3 and some mystery value "x", then add 7 (we must follow the order of operations! See the early blog post to review the order of operations).  This is a pretty easy expression to evaluate.

Let's find the value of the expression if we let x = 4.
When x = 4, we can rewrite the statement and instead of writing the "x" we can replace it with 4.
So the expression would become:

3(4) + 7

Note the use of parentheses to remind us that the expression was 3 times x, or now, 3 times 4.
The value of this expression is now 12 +7, which is 19.

Evaluating expressions becomes trickier when the expressions themselves have more terms, involve exponents, or tricky orders of operations.

For some basic review, check out Khan Academy's site (click here for video) and (click here for practice problems).

For some fancy, challenging problems, check out this website (click here). The answers are provided, so you can check your work.

This site walks you through the evaluation process step-by-step. (click here) If you go to this link (click here) you can practice some really challenging problems. They call it a "work out", and it will wear your brain out. But it's like any other muscle... the more you exercise it, the stronger it will become. Try to get 8 right in a row before you move on to a different topic.

More sign rules!

More advanced sign rules for addition and subtraction...

Often in an math problem with addition and subtraction of positive and negative numbers the signs and operators can become confusing. I'm going to teach you a fast way to take (what I call) 'double signs' and simplify to a single sign so the problem will be more manageable.

Let's look at this example:

-4 + (-3)   here we have a negative 4 being added to a negative 3. There are several ways to think about this type of problem, but the easiest way for some students is to use multiplication rules to simplify the double signs.  We know the multiplication rules are:

(+)(+) = +         When there are two signs next to each other in addition or subtraction
(-)(-) = +           we can remember these rules and use them to pick just ONE sign to use
(-)(+) = -           when we rewrite the problem.

So -4 + (-3)  has a positive (or plus) sign next to a negative sign.  A negative and a positive sign simplify to a negative sign.

The problem can be rewritten as -4 - 3.  Using the rules we learned in an early blog post, these two numbers have the same signs and we will add the numbers and keep the sign.  So the answer is -7.

This website (click here) has a nice explanation of these types of problems and shows several different examples broken down for you.

This website (click here) has endless practice problems for you to try. You can enter your answer into their website and they will check your work right away. Check it out - it's very useful!

Here is a worksheet that you can try to make sure that you are beginning to grasp the concept. The answers are included after the worksheet (click here)

And as always, if you need any additional help, stop by the Learning Lab department and make a tutoring appointment!

Sign Rules!

Math isn't just about numbers anymore!  It's about numbers, and letters (also called variables), and and SIGNS.

Knowing what the sign rules are and when to use them is key to succeeding in your math class.  In this post we will focus on adding and subtracting sign numbers.

Here is a link to a video explaining the sign rules for addition and subtraction of sign numbers (click here!)

Watch the video and then try these practice problems:

1.)   -5 + 6 =


2.)   22 - 10 =


3.)  -14 - 23 =


4.)  6 - 6 =


5.)  -6 - 6 =


6.) 32 - 12 =


7.) -9 + 4


8.) -12 + 42 =



9.) 50 - 20 =


10.) -55 + 25 =

Answers:
1.) 1       2.) 12      3.) -37    4.) 0    5.) -12   6.) 20    7.) -5    8.) 30    9.) 30    10.) -30

You can find LOTS of practice problems on the Community College of Philadelphia's Math Department page. Click here and work through the first three and a half pages of signed number problems.

Welcome to our blog!

Welcome to the Learning Lab Department's Math Review Blog!  This is a site where we will gather all of the best on-line math materials for you to use as you work your way through Math 016 and 017.

Whether you find yourself stuck, or you just want to find more practice problems to improve your skills, visit this blog and search through the posts to find the materials you're looking for.

Often, the help that you need to tackle a hard concept or brush up on rusty skills is available on-line. You just have to know where to look! We have found the best resources available and will walk you through how to use them so you can improve your math skills, build confidence, and pass your math class.

If you're not quite sure where to begin, your are welcome to find one of our talented math tutors at any of our college campuses.

In-person tutoring for math classes is located:

Main Campus: B2-36 (from 9am to 4pm Monday through Friday) and B1-28 (from 4pm until late evening - the final tutoring time depends on the semester and day). You can call 215-751-8481 if you have questions.

Northwest Campus: Room 121 (by the computer lab). Tutoring at the Northwest is done on a drop-in basis. Math tutoring is usually held from 10am until 6pm on Monday through Friday and from 10am until 3pm on Saturdays. You can call 215-496-6020 to call and confirm the schedule.

Northeast Campus: Learning Commons. Tutoring at the Northeast usually runs from 9am until 7pm Mondays through Fridays with some Saturday hours available. You can call 215-972-6236 to make an appointment and/or check the tutoring hours.

West Campus: Learning Commons. The schedule varies, but there is usually tutoring several days a week. Call 267-299-5848 to ask about the tutoring schedule.


We also have Math Workshops available at a variety of campuses through the semester. Check out this link (click here) to find when the workshops are happening near you (click on the "workshop" link and then on the "math workshop" link to find the complete schedule).

If you ever have trouble finding anything on this blog or you need help with your math course you can email me at mfuller@ccp.edu.

Good luck!
Megan Fuller