Algebra Terms: 'Evaluate' and 'Simplify'

Date: 7/30/96 at 14:38:11
From: Ralph Ware
Subject: Algebra Terms

Dear Dr. Math:

I am a 43-year-old man entering college this fall for the first 
time.  One of my required courses is Elementary Algebra. It has been 
25 years since I've seen an algebra problem and I didn't exactly set 
the world on fire with my mathematical ability back then, even though 
I passed both Algebra I and II in high school.  
   
The college has sent me a sample list of problems I must be able 
to work in order to remain enrolled in the course. Some of the 
mathematical skills I learned in elementary and high school remain 
familiar because I use them in solving every day problems.  But I am 
unfamiliar with some of the terms used in the list.  I have found the 
"Ask Dr. Math" portion of the Math Forum to be very helpful in getting 
back into the swing of things. 

Even though I am a lot older than most of the students who write 
you, I hope that you will help me with my questions, too.

1. Evaluate: 3-{2-3(2-7)}   What does "Evaluate" mean?

2. Simplify: -3x{7-(3x-5)}  What does "Simplify" mean?

Any help you can give me will be greatly appreciated.  Thank you.

Ralph Ware

Date: 7/30/96 at 15:33:7
From: Doctor Robert
Subject: Re: Algebra Terms

Evaluate means to "find the value" of.  It is similar to the old 
question, "what is 10 plus 4?"  Evaluate (10+4) should evoke the 
same answer. To answer your particular question I'll work it out step 
by step:

3-(2-3(2-7)) = 3-(2-3(-5))=3-(2+15) = 3-17 = -14.  When evaluating an 
expression, you must pay attention to the ORDER OF OPERATIONS.

Simplify means to make more simple.  In your example you cannot 
evaluate the expression for you are not told the value of x.  
However, you can, using the order of operations, simplify the 
expression.

-3x(7-(3x-5)) = -3x(7 - 3x + 5) = -3x(12-3x) = -36x + 9x^2.  

I hope that this helps.

-Doctor Robert,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 7/30/96 at 15:42:15
From: Doctor Pete
Subject: Re: Algebra Terms

"Evaluate" means that a particular value is desired for the answer. In 
other words, they want a single number.  "Simplify," on the other 
hand, means "to make simpler," or to reduce the number of symbols used 
while retaining equality.  So for "simplify," they don't want a single 
number, but instead they're looking for something like "4x" or "3x+2" 
- which is as short and sweet as you can make things.

So to "evaluate" the first problem, we start from the inside and work 
our way outwards:

     3-{2-3(2-7)} = 3-{2-3(-5)}
                  = 3-{2-(-15)}
                  = 3-{2+15}
                  = 3-17
                  = -14 .

Notice the answer is a number (in this case, it is negative), not an
expression with a variable in it.  

For the second problem, we have

     -3x{7-(3x-5)} = -3x{7-3x+5}   (distribute the minus sign)
                   = -3x{12-3x}
                       2
                   = 9x -36x .     <-- this is "nine x squared minus
                                       thirty-six x"

Now, although this is probably the "simplest" form, it isn't the way I 
would want to write it.  I'd rather say this is

                   = 9x(x-4) ,

because it tells me more.  It says, "Well, if x=4, the expression is 
0." But in general, "simple" means fewer parentheses and groupings, as 
well as fewer operations (plus, minus, multiply, divide) used.

However, in general, it doesn't really matter whether somebody says
"simplify" or "evaluate."  They just want you to make the expression 
as short or compact as possible.  The technical difference is that 
"simplify" usually refers to an expression with variables, where you 
(usually) can't get a single number out of it, while you can with 
"evaluate."

-Doctor Pete,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 8/1/96 at 16:59:29
From: Doctor Mike
Subject: Re: Algebra Terms

Hello Mr. Ware,
  
Your questions are appropriate.  We cover whatever COULD BE in grades
K through 12.
  
College teachers don't mind older students.  In fact, students with 
more maturity who take the subject seriously may even have an 
advantage, since most teachers also tend to take their subject 
seriously.  Have fun. 
  
If what I'm writing below makes a lot of sense, and what you learned 
from High School starts coming back to you more and more, then I am 
glad to have helped with that.  But if it is still pretty confusing 
then go ahead and take a lower course that will include a lot more 
review.  I'm pretty sure that these questions "in order to stay 
enrolled in the course" are just an informal "placement test" to make 
sure you are placed in the  right course to correspond to your 
background.  They want you to succeed.
  
Let's take "Evaluate" first.  That just means to find out what that 
long expression is equal to.  You have to use things you learned in 
algebra, like [1] Always do the operation inside the parentheses 
first, [2] A negative number times a negative number is a positive 
number, [3] The distributive law for "multiplying out" stuff, like 
a*(b+c) = a*b + a*c , etc.  There are non-obvious forms of that rule, 
like the following:

      a*(b-c) = a*b - a*c
     -1*(b+c) = -b - c
     -1*(b-c) = -b + c
       -(b-c) = -b + c      etc. , etc., etc.

I'll give you a step-by-step for your example so you get the picture:

   (a) (2-7) gives -5
   (b)  -3 times -5 from step (a) gives 15
   (c)  2 plus 15 from step (b) gives 17
   (d)  3 minus 17 from step (c) gives -14

so -14 is the numerical value of that expression; we "Evaluated" it. 
  
Now for "Simplify".  This is not as straightforward as the other since
various people may have different ideas of what is more simple than 
what else.  But everybody agrees that -3x{7-(3x-5)} could be made 
simpler. Keep in mind here is that "x" represents some number. You do 
not know what that number is, but you are guaranteed that you can do 
anything with "x" that you can do with ANY number.  

Here is a step-by-step for what I would do for this one : 

   (A)  -(3x-5) is the same as -3x + 5 from one of the versions of
        the distributive law.
   (B)  So, what is within the braces is 7 - 3x + 5 .
   (C)  If you want, you can think of this as 7 + (-3x) + 5 to show 
        that this is just 3 numbers added together, and you know that 
        numbers can be added in any order, so re-write it as 
        7 + 5 + (-3x) .
   (D)  Simplify the (C) expression to just 12 - 3x .
   (E)  So now the original expression amounts to -3x times 12-3x .
   (F)  How could that be simplified?  Here is where we get into the
        nebulous area of interpretation.  Since A*B = (-A)*(-B) it
        follows that -3x times 12-3x is the same as 3x times 3x-12 .
        That looks simpler to me since the minus sign is gone from 3x. 
   (G)  Some would think (3x)*(3x-12) is a nice simple factored form,
        whereas others would prefer 9x^2 -36x in multiplied out form.
         NOTE: I'm using x^2 for x squared.
   (H)  People who preferred factored results might want to go the 
        other direction and factor out 9x to get (9x)*(x-4).  This
        is what I would prefer, but if your prof has different ideas
        about it, keep in mind who will grade your exams!  
  
I hope this is exactly what you needed.  If not please write back. 
  
-Doctor Mike,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/