Lesson HOW TO WORK WITH + AND - for multiplication and division

This Lesson (HOW TO WORK WITH + AND - for multiplication and division) was created by by longjonsilver(2297)

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About longjonsilver: I have a new job in September, teaching

There are 2 parts to understanding how to use signs correctly in maths. These are when dealing with ADDITION and SUBTRACTION and then with MULTIPLICATION and DIVISION. I cannot offer proofs of what I say or derivations, as basically I do not know them. They are also beyond the level of most courses. Also, they are quite boring, in my opinion. Sorry purists :-). However some other sites do offer potentially helpful examples, such as http://www.tpub.com/math1/4b.htm. However, for me they do my head in.

Multiplication and Division
MULTIPLICATION

This is the easier part to do first, so here goes with some examples:

(+2)*(+3) = +6
(+2)*(-3) = -6
(-2)*(+3) = -6
(-2)*(-3) = +6
(+5)*(+4) = +20
(+5)*(-4) = -20
(-5)*(+4) = -20
(-5)*(-4) = +20

OK, basically multiplipying (or dividing) 2 numbers then following these rules:

if the 2 numbers have the SAME sign, then the answer is +
if the 2 numbers have the DIFFERENT sign, then the answer is -

so, %28-2x%29%2A%28-3y%29+=+%2B6xy

.

Get into the habit of breaking the question down. Do not think of -2x as one thing and -3y as the other. Instead think of the signs first... - and - give a +...so that is done. Next the numbers, 2x3 is 6. That is done. Then x... no other x terms to multiply with, so it is still x and then similarly the y..it has no other y terms to multiply with, so it remains y.

EXAMPLE:

3ab * -2a = -6a%5E2b

So, I look at this as follows:
+ and - --> -
3x2 --> 6
axa --> a%5E2

b --> b

How about (2ab)*(-4b)*(-3ab)?

Remember, the sign rules above apply to ONLY 2 signs at a time, so initially do the above in 2 steps:

%282ab%29%2A%28-4b%29+=+-8ab%5E2

then

%28-8ab%5E2%29%2A%28-3ab%29+=+24a%5E2b%5E3

Once you are happy with multiplication of many terms, you can do the signs in your head in one go: so for the above example, I would say + and - (from the 2 and the -4) is a -. Then this - and the - (from the -3) is a +.

DIVISION

This follows the exact same rule as multiplication, as you would expect since they are two sides of the same thing (ie 2*4=8 or 8/2=4 --> opposite processes).

So, I shall just do some examples:

(+12)/(+3) = +4
(+12)/(-3) = -4
(-12)/(+3) = -4
(-12)/(-3) = +4

And algebraic examples:

%28-12a%29%2F%28-4%29+=+3a

. Again, do the 2 signs. Then do the 12 and 4 then do the letters, in this case we only have one.

%286a%5E3%29%2F%28-12ab%29+=+%28-a%5E2%29%2F2b

. Again, do the signs, then the numbers, then the a-terms and then the b-term.

How about multiplication and division? Which should i do first? It doesn't matter, since as I said earlier, they are 2 sides of the same coin.

EXAMPLE: Simplify %28%286a%5E2bc%5E3%29%2A%28-4a%5E2c%29%29%2F%28-2a%5E3b%5E3c%5E2%29

%28%286a%5E2bc%5E3%29%2A%28-4a%5E2c%29%29%2F%28-2a%5E3b%5E3c%5E2%29

You can do any of this in any order, as the terms are all connected by either multiplication or divide. Realistically, to stop yourself getting in a muddle, you would probably pick the multiplication on the top first, to get that into one combined term, as follows:

%28-24a%5E4bc%5E4%29%2F%28-2a%5E3b%5E3c%5E2%29

%28-24a%5E4bc%5E4%29%2F%28-2a%5E3b%5E3c%5E2%29

and now we can do the division, to give 12ac%5E2%2Fb%5E2

In all this, if there is no sign written, then it is assumed to be a +, so the last answer is in fact + 12ac%5E2%2Fb%5E2

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