Thread: The use of division in simplifying.

I'm having a hard time trying to understand the use of how to deal with division in simplifying.

For instance with multiplication if you had (-6n) X 3n you take the numbers and the letters seperately so -6 X 3 woudl be -18 and n X n would be n squared.

But with 3x / 12 3 divided by 12 would be 0.25 and x divided by 12 would be what? the answer given is x/4 and I can kind of see this if i think about it as a rational number and then cancel both parts by 3 to give x/4 but this is just guess work on my part as there's no mention of this in the chapter I'm working on.

Am I thinking about this right or are there some short cut rules that I'm missing?

Also in 8x / 4x with is the answer 2 and not 2x?

Thanks.

Confusing, isn't it?

Ah, how about like this:
You are to divide only aples by apples, oranges by oranges, etc.
Numbers by numbers, letters by letters.

3x / 12

Numbers by numbers:
3 / 12
= 1 / 4

Letters by letters:
x /....what? there is no x in the denomitor.
So you cannot divide x.
so leave it there in the numerator.

Hence,
3x / 12
= 1*x / 4
= x/4 -------------answer.

-------------------------------------------

8x / 4x

Numbers by numbers:
8 / 4 = 2

Letters by letters:
x / x = 1

So,
8x / 4x
= 2*1
= 2 ---------------answer.

I'm still not sure I understand. You seemed to treat the first as a fraction that needed simplifying but in the second you seemed to do something else. In the second example I was thinking of it again terms of a fraction meaning the 8 and 4 would reduce to a 2 and 1 and the x's would cancel each other out meaning you would be left with 2/1 which can be further simplified as 2.

As I said i'm taking a leap that these should be considered as fractions in order to simplify them. Is this right?

Assuming it is i'm also having a problem understanding at what point in the simplifing process the idea of dividing by a negative number in either the lower or upper part of the fraction alters the other part. For instance 9e squared /(-12e cubed) is it that as soon as I simplify both numbers that 9 becomes -3? and is it just convention that mean the answer is expressed with both the numerator and denominator as positive number with a single minus sign before the whole fraction or is there a deeper reason for this?

Another exampel of this is (-a squared) / (-a) squared. here you have -a x -a / -a x -a. If I follow the normal rules here they all cancel each other leaving me with 0. however the answer is -1?

Again thanks for the help.

Apples divided by apples, oranges divided by oranges.

9 = 3x3, 12 = 3x4. There's a factor of 3 in both of them so you can cancel the 3 out. Putting something on the bottom of a fraction is like saying "divided by" that number. So it's like saying:



which is like




Dividing is "undoing" times, so times 3 is "undone" by "divided by 3".

(Going from 9/(-12) to -3, what you've done is subtracted which ain't right.)

So you got 3 on the top (one left out of the two) and on the bottom you got -4 and e, so you end up as:

is the answer I came to too. But the answer I have in the back of the book has a minus sign in the middle of the franction and then positve numbers on both the numerator and denominator implying the whole fraction is negative. Is this wrong?

Also i've just been reading up on this and a different method i've found is divide the coefficents and then and subtract the exponents. But when I apply this to the first problem I posted I end up with 2x and not 2 which is the answer I've been given.





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Originally Posted by Alyosha

But with 3x / 12 3 divided by 12 would be 0.25 and x divided by 12 would be what? the answer given is x/4 and I can kind of see this if i think about it as a rational number and then cancel both parts by 3 to give x/4 but this is just guess work on my part as there's no mention of this in the chapter I'm working on.

Also in 8x / 4x with is the answer 2 and not 2x?

The biggest problem I see is that you are looking for shortcut rules. don't do that. Do what is right, first.

The next biggest problem is that no one seems to be talking about the "Identity Element" or the "Reciprocal". You should know this...



, if x is NOT zero.



To your problems:



There is just no confusion forging through in this way.

Similarly,



And again, I just don't see the confusion.

I blame your teachers for showing you how to muddle through rather than insisting that you understand what it is you are doing. You should have studied properties such as the Associative Property of Multiplication. These things may seem dry and arid when first encountered, but now is the time to drink deeply from their well of rigorous water. Let the camels muddle through the sand.

Originally Posted by Alyosha

is the answer I came to too. But the answer I have in the back of the book has a minus sign in the middle of the franction and then positve numbers on both the numerator and denominator implying the whole fraction is negative. Is this wrong?

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No it's not wrong. Sorry, I glossed over the minus.



It honestly doesn't matter where you put that minus sign, as long at it's over at the left hand side ...

but wouldn't


mean that both numbers were negative so that -1 divided by -1 would be +1?

I'm not really sure I understand what it is you're doing here



are you creating the 4 * 3 with the common factor or finding a number that is equal to the coefficent so that you can create a seperate fraction will equal 1? meaning that if you had 6 * x you would have 6 * 2?



here I can you've split the 8x to a common factor sum of 4 * 2x but I can't see why you've then multiplied both side of the fraction by 2.

No, the minus sign applies to the whole thing.



Minus divided by plus = minus.

Plus divided by minus = plus.

Minus divided by minus = plus.

Now

Originally Posted by Alyosha

I'm not really sure I understand what it is you're doing here

I don't want to say that you are overthinking it, but you are thinking about it in a NONuseful way. It is NOT as complicated as you seem to be making it.

The expressions I have written represent a progression of the process from left to right. Take each pair of smaller expressions (one equal sign and the two expressions surrounding it) and see how it works.



The ONLY thing that happened here is that 12 = 3*4. That's it. It is neither shocking nor difficult. I also rewrote the numerator with an explicit multiplication symbol, just to make it look more like the denominator. This is only cosmetic, not magic mathematics.

Now the next one.



Here we used properties of fractions. What is the rule for multiplying fractions. "Multiply numerators and multiply denominators". Look at this until you see that this rule has been applied in reverse. Given the right hand side, can you get the left hand side? This should also ring some bells concerning the cummutative and associative properties of multiplication.

Next one.



Here, we applied the principle of the multiplicative inverse. 3/3 = 1. That's all that occurred.

Finally



We applied the property of multiplicative identity. Multiply things by 1 and nothing happens?

The whole process is the application of the most fundamental principles. If you think it is new material, then you have not learned the truth. All the stuff you have been learning since 1st or 2nd grade should be brought to bear on such problems. I am not making fun of you, I am just emphasizing that this is NOT new material. You already know the pieces of the puzzle. Just put them together. None of it should be a surprise.

You do the other one. Explain how to get across each equal sign.

In the second example if I'm using the inverse of the idea of multiplying the numerator and denominator by the same number, mean dividing them by the same number, or common factor, which would be in this case 2. Wouldn't that leave 2 * 2x that equals 4x? as you've only seemed to divide the numerator and left the denominator alone.

A common factor does not have to be a number, it may also be a variable, in this case .

yes, indeed it has a common factor of 2. And it's got another common factor of 2. There's two common factors of 2. This is because and . So you can look upon your as being

If it makes it easier, you can also write it as:


because multiplying something by 1 doesn't make any difference to it.

So by the rule of multiplying fractions, but backwards):




As you can see, all but the first bit equal 1, leaving that last lonely little 2 all out on its own all unloved.

Originally Posted by Alyosha

In the second example if I'm using the inverse of the idea of multiplying the numerator and denominator by the same number, mean dividing them by the same number, or common factor, which would be in this case 2. Wouldn't that leave 2 * 2x that equals 4x? as you've only seemed to divide the numerator and left the denominator alone.

You are NOT multiplying numerator or denominator by anything.

You are NOT dividing by the same number anywhere.

You SHOULD BE applying the properties with which you should be familiar.

If you are NOT familiar with these properties, it is likely that you are in the wrong class and you should back up and get a better background.

Time for a Placement Exam!

1) What property is demonstrated here?

(a*b)*c = a*(b*c)

2) What property is demonstrated here?

a*0 = 0

3) What property is demonstrated here?

b*c = c*b

4) What property is demonstrated here?

a*1 = a

5) Find the Prime Factors of 12.

6) What property is demonstrated here? (For a <> 0)

a*(1/a) = 1

Six quick problems that should be second nature by now. Get 100% or I will be discouraged.

1) The idea that you can multiplying things in any order

2) that anything times 0 = 0

3) same as 1

4) that anything *1 = 1

5) 2 X 2 x 3

6) assuming this (For a <> 0) means a is greater or lesser than 0, it's that any number multiplied by it's reciprocal is 1.

Originally Posted by Alyosha

1) The idea that you can multiplying things in any order

2) that anything times 0 = 0

3) same as 1

4) that anything *1 = 1

5) 2 X 2 x 3

6) assuming this (For a <> 0) means a is greater or lesser than 0, it's that any number multiplied by it's reciprocal is 1.

Not bad. I am officially not discouraged. They do have names. You should learn them.

1) Associateive Property of Multiplication
2) Zero Property of Multiplication
3) Commutative Property of Multiplication
4) Identity Property of Multiplication
5) You totally got this one. Good work.
6) Inverse Property of Multiplication

"<>" is code for "Not Equal To". It's a little clunky, but it is in rather general usage.

Okay, now that we have these properties firmly in our minds, we can drag through one of these simplifications.

We are presented with .

The fundamental idea is to find ways to utilize the Inverse Property of multiplication. In this way, we can change very complicated expression in to much simpler expressions.

First, then, we use the Associative Property of Multiplication in the Numerator. Rather than group everything together, we'll regroup a chunk out.



Nothing else happened. Just that one property. I picked this particular application of a property because it left in the numerator something tantalizingly close to what is in the denominator. I may have played around with the prime factorization a bit until I found something that looked right.

Second, we're going to use the Associative Property of Multiplication again.



Before, you said I multiplied everything by 2. This simply is not the case. I have only changed the order of things a bit. Rather than doing multiplication in the numerator first, this expression may indicate doing something with the other two factors first.

Now, we're going to hope that x is NOT zero (and by extension that 4x is not zero) so that we can use the Inverse Property of Multiplication. This allows us to see that a/a = 1 as long as a is not zero. With this insight, we can deal with the fraction.



Notice, please, that the application of the fundamental inverse principle is ALL that occurred.

One last thing, the Identity Property of Multiplication



Again, just one step at a time. Look long and hard at each step. Don't try to complicate it. You need to get good at this so you can do it quickly.