Simplifying Answers

Date: Sat, 17 Dec 1994 20:33:07 AST
Comments: NB*net - New Brunswick's Regional Network 
   1-800-561-4459
From: Richard Seguin

How do you answer Questions like this?

[(m "to the power of -3" n "to the power of -2" p) (m "to the power of 2" n)
(M n "to the power of 2"]"to the power of 5". 
 
It's kind of hard to write those kinds of questions but if you could try to
help me it would be appreciated! 

Date: Sat, 17 Dec 1994 20:25:42 -0500 (EST)
From: Dr. Sydney
Subject: Re: your mail

Dear Richard,

        Hello!  Of course, we'd love to help with this kind of question.
Notation can get kind of hairy on email, but we have to work with it.  
I'm not quite sure what you are asking because of notational problems.

We usually notate taking a number to a power with a ^.  So, 2 to the 3rd
power would be 2^3.

Maybe you could write us back using this notation or even standard 
notation

        3  
(like  2     for 2 to the third power)

I look forward to hearing from you again soon.

Sydney, Dr. "math rocks my world" foster

Date: Sun, 18 Dec 1994 10:25:52 AST
From: Richard Seguin

Sorry About the problem. I am new at writing math on my computer. 
Anyways, 

The Question is:

(P^2Q^-4)^2(P^3Q^5)^-3  <---- LOOKS A LITTLE STRANGE 
BUT THATS WHAT YOU TOLD ME TO DO. 

The second question I had was:

[(m^-3n^-2p)(m^2n)(mn^2]^5  <-- I don't understand what comes first, 
and what to do first. 

Date: Sun, 18 Dec 1994 11:32:52 -0500 (EST)
From: Dr. Sydney
Subject: Re: your mail

Dear Richard:

        I'm glad you wrote back.  Yes, this notation is a little strange,
but it works!  Just to clarify, are you asking in the first question how to
simplify 
          2  -4                          3  5
         p  q  all to the power 2 times p  q  all to the power of -3 ?

(It's not p to the power 2q to the power -4, right?)  Assuming that you 
are asking the above stated question, let's try and figure something out.
Before starting, though, it is important to remember the rules of
mulitplication with exponents:

        1) If you are multiplying two numbers together that are raised to
different powers from the same number, all you do is add the exponents
together.  In math language:   b  c   (b+c)
                              a  a  = a

You can see why this is true if you look at an example.  The above rule
tells us   2  3    5
          2  2  = 2  
                              2                  3
Does that make sense?  Well, 2 is just 2*2 and 2 is just 2*2*2 (the * is
mulitplication).  So if we multiply the two numbers together we get:
                                                5
2*2*2*2*2, right?  And that's just the same as 2.

        2) If you have a number that is raised to a certain power already and
you raise that number to another power, that is the same as raising the
original number to the product of the two powers.  In math language:
                             c   bc
                        (a^b) = a

 Again, if you think about specific examples this will make sense.  For
                                     3    3*2
example, this rule tells us that (2^2) = 2 ....let's see if it makes sense.

Well, 2^2 is just 2*2, and if we raise that to the 3rd power, we get
(2*2)*(2*2)*(2*2) = 2*2*2*2*2*2 which is just 2^6 or 2^(3*2), right?

Okay, so now that we have the basic rules down, we are ready to attack 
your problems:

I'll help you with the first one, and maybe you can try the second one on
your own. If you write back with the answer you get I can tell you if it is
right.

Okay, so the problem is to reduce:

[(p^2)(q^4)]^2 [(p^3)(q^5)]^-3

First, let's reduce the first part: 

[(p^2)(q^4)]^2

We'll use the second rule here:  

So, multiply the exponents on the inside of the brackets by 2 to get:

(p^4)(q^8)

Now simplify the second part:

[(p^3)(q^5)]^-3

Again, using the second rule, multiply the exponents on the inside of the
brackets by -3 to get:

(p^-9)(q^-15)

Now, we've simplified both parts of the expression. We just need to bring
them together and use the first rule to completely simplify the expression.
So our problem now is to simplify:

(p^4)(q^8)(p^-9)(q^-15)

By the first rule, we just add the exponents with similar bases together.

So, (p^4)(p^-9) = p^-5           and (q^8)(q^-15) = q^-7

So, the simplified answer is just:

   (p^-5)(q^-7)

Did that make sense to you?  Why don't you try the next one and write 
back with your answer (include the question in your email, too, please!).  
If you have any questions on this please do write back.

--Sydney

Date: Mon, 19 Dec 1994 07:28:49 AST
From: Richard Seguin
Subject: rseguin@nbnet.nb.ca

Ok, The question was:

  2 -4 2    3 5 -3
(p q  )  (p q ) 

If you don't understand that, then:

(p^2q^-4)^2 (p^3q^5)^-3

Ok, for the first part I simplified the following:
(p^4 q^-8) 

The second part I simplified the following:
(p^-9q^-15)

Then I added my exponents

(p^4q^-8) (p^-9q^-15)

= (p^5q^-23)  <----- That was my answer..  Did I do it right?

Now, if I had to get an answer how would I do the next step?

Date: Mon, 19 Dec 1994 13:42:52 -0500 (EST)
From: "Michael W. S. Morton"

Hello there!

        I'm another one of the Math Doctors here to help ya out.  Let's 
take a look at what ya got:

> = (p^5q^-23)  <----- That was my answer..  Did I do it right?

Your work and answer all looked good to me!
The steps that you took were the exact same ones I would!

> Now, if I had to get an answer how would I do the next step?

This is as far as you can go with what you were originally given, i.e. 
just to reduce to a simpler form. But, say you had what this expression 
was equal to (some number), then what would you do? (I hope this is 
your question :)

Let's first take a look at a simple example, say:
p^2 q = 5               So, the 5 is what the expression equals.

How do you solve this?  Let's first move the p^2 to the other side, so:
     5
q = ---     The answer to this equation is a whole line of points!
    p^2     It is just the line like y=5x^-2.

So, the answer to this is a whole line of points, what about your original 
equation:

p^5q^-23 = 5     Where 5 is given.

It is also just some sort of 'line', where each point on the line is an 
answer for (p,q) satisfying this equation.  You can even graph this 
(hopefully using a computer or a graphic calculator :)

I hope that helps ya... Let us know if I missed yer question or if you 
have another!!

                                        -MORTON, doctor of sorts

Date: Mon, 19 Dec 1994 14:40:44 -0500 (EST)
From: Dr. Sydney
Subject: Re: rseguin@nbnet.nb.ca

Dear Richard, 

        I was just looking over your original question and my response to
your question, and I realized that I accidentally miscopied the question
when I was answering it later in my first message.  So, my first message
showed you how to reduce :  [(p^2)(q^4)]^2 times [(p^3)(q^5)]^-3

Sorry for the mistake, I hope it didn't confuse you.  At any rate, it gave
you a chance to practice simplifying these expressions, right?  (Come 
on, humor me!)

Everything in your work looks good, except one minor detail.  At the end,
when you add exponents, you should have gotten (p^-5)(q^-23).  That is, 
you should have gotten p raised to a NEGATIVE 5, rather than a positive 
5 because 4 - 9 = -5.  So, your strategy was excellent -- just a little
arithmetic error.  

Hope that makes sense.  Write back if you have any more problems.

Sydney