simplify complex fractions:

- anonymous

simplify complex fractions:

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- amistre64

complex as in with the imaginary "i"?

- anonymous

all you need to know is i^2 = -1

- anonymous

no mutiplying and dividing rational expressions
\[(x^3y^2z)/(a^2b^2)/(a^3x^2y)/(b^2)\]

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## More answers

- anonymous

where the slash is its written like a fraction

- anonymous

So what is over what here?

- anonymous

stack the parenthisis up ontop of each other where it looks like a fraction with things. idk how to punch it in on here were it looks like it does on my worksheet

- amistre64

you can use // to mean the main divider bar between them, or some other version like */* just explain the symbol :)

- anonymous

From what it looks like, I would assume it's like a rational expression over another. Correct?

- amistre64

the b^2.s cancel out

- anonymous

yes rational expressions

- amistre64

basic strategy as with all fraction on fraction math is to turn the bottom one upside down and multiply across...
1/5 // 6/3 = 1/5 * 3/6 = 3/30

- anonymous

but when its all letters and exponets how do you do it

- amistre64

you mean the multiply part? or the flipping part?

- anonymous

the multiply part. flipping i get now but idk what to do with the letters

- amistre64

make it easier on yourself and split them up...
a^4 means aaaa
c^2 mean cc
r^7 means rrrrrrr

- amistre64

now, x^3 times x^4 means
xxx xxxx = xxxxxxx = x^7

- amistre64

c^4 times c^5 means
cccc ccccc = ccccccccc = c^9

- anonymous

yea i get the x^4 means xxxx but what about x^3y^2z

- amistre64

xxx yy z is all that is

- amistre64

squish em all together, then count them back out again....

- anonymous

what your confusing me

- anonymous

a/x = a* 1/x, therefore \[(x ^{3}y ^{2}z)/(a ^{2}b ^{2})/(a ^{2} x ^{2} y)/(b ^{2})] =
\[(x ^{3} y ^{2} z)/(a ^{2} b2) * 1/(a ^{2} x ^{2} y)/(b ^{2})] =
\[(x ^{3} y ^{2} z)/(a ^{2} b ^{2})*(b^{2})/(a ^{2} x ^{2} y)\]

- amistre64

xxx yy z bb
-------- x ------
aa bb aaa xx y

- amistre64

anything that is the same on the top and bottom can be tossed out, it just means it equals 1.
x y z 1 x y z xyz
----- x ---- = ------ = ----
aa aaa aaaaa a^5

- anonymous

still not makin a whole lota sense

- amistre64

which part is not making sense?

- amistre64

i flipped it, you said thatpart you understood.... whats left?

- amistre64

Does:
xx y bb
------ = 1?
xx y bb

- anonymous

the whole thing. its a fraction placed ontop on a fraction saying simplify. like am i supposed to add the exponets together, flip them and multiply, or flip and divide

- amistre64

flip the bottom, thats always the first step.
split the exponents up so you can see what you got
whatever is the same from top to bottom cancels out to 1
squish the rest together and number them again with an exponent.

- amistre64

lets keep it simpler, how would I start this?
x^2/y^3 // y/x^2

- anonymous

flip it and take away the x^2 and just leave it as xy^3/xy i think

- amistre64

one step at a time, dont rush ahead... lets flip the bottom
x^2 x^2
---- * ---- now what do I do?
y^3 y

- anonymous

the x^2
----
y part

- anonymous

oh wait
y^3 x y
-------
x^2 x^2

- amistre64

you let me worry about making it look good on the screen, just tell me what we do next...

- amistre64

I already flipped it,
x^2 x^2
---- * ---- now what do I do?
y^3 y

- anonymous

i want to say crossmultiply

- anonymous

but thats usually not with letter

- amistre64

we do that if there is an (=) between them, there is no (=) here so we dont "cross" multilply, we just multiply straight across

- anonymous

ok just add the exponets together

- amistre64

-------------->>
x^2 x^2 x^2 x^2
---- * ---- = ----------
y^3 y y^3 y
-------------->>
does this look right to you?

- anonymous

not really shouldnt it be x^3/y^4

- amistre64

not yet.... but does the setup look right? did I multiply it across correctly?

- anonymous

yea i think.

- amistre64

now lets add exponents to get the final result
x^2 x^2 x^(2+2) x^4
------- = --------- = ----
y^2 y y^(2+1) y^4

- amistre64

2+1 = 3... sorry, forgot how to add :)

- anonymous

that looks more like it

- amistre64

so lets try your original problem and see if we can step thru it ok?

- anonymous

ok

- amistre64

x^3 y^2 z
---------
a^2 b^2
---------------
a^3 x^2 y
--------
b^2
our first step is to do what?

- anonymous

im working it on a sheet of paper and the first thing i did was put the bottom fraction beside it and put a division sign between them

- amistre64

ok....
x^3 y^2 z a^3 x^2 y
--------- / ----------
a^2 b^2 b^2
now what?

- anonymous

um flip the entire thing upside down or add the exponets

- amistre64

only flip the right side fraction... not the "whole" problem....JUST that right side gets flipped. right?

- anonymous

ok

- amistre64

just like this....
x^3 y^2 z b^2
--------- ----------
a^2 b^2 a^3 x^2 y

- anonymous

yea then add the exponets together

- amistre64

what do you mean by add the exponents together? that is not really something that needs to be done just yet. Look at the top and the bottom of this oversized fraction and see if we can cross out stuff that looks the same.

- amistre64

lets "line" things up from top to bottom....are we allowed to do that?

- anonymous

line things up idk what your talking about

- amistre64

I am going to move the stuff around so that it looks better.
Like this.....
x^3 y^2 z b^2
--------------------
x^2 y a^2 a^3 b^2

- amistre64

what can you see that we can "get rid of" that is the same from the top and the bottom?

- anonymous

um on the bottom combine the a^2 and a^3 together to get a^5

- amistre64

good we can do that...
x^3 y^2 z b^2
--------------------
x^2 y a^5 b^2

- anonymous

can we also combine the x^3 and x^2 or not since there on seperate parts of the equation

- amistre64

we cant "add" them together but watch this:
xx x
---- = what?
xx

- anonymous

5

- amistre64

no....not quite.
do you remember that anything when placed over itself is equal to 1?
2
-- = 1
2
6m
--- = 1
6m
xx
-- = 1
xx

- anonymous

i think

- amistre64

this is a fundamental concept in math.
whenever we "divide" a number by itself we get an answer of 1
1
---
8 | 8
right?

- anonymous

i guess but i dont particularly see how that would work in this type of problem

- amistre64

all fractions are is division.
what is 1 divided by 2 = 1/2
3 divided by 5 = 3/5
15 divided by 5 = 15/5
all fractions are division......

- amistre64

when we divide a number by itself we get 1
9 divided by 9 = 9/9 = 1

- anonymous

but in this case were dividing x^3/ x^2

- amistre64

exactly :) which is why I like to split it up so that you can "see" what is going on with exponents.
x x x xx x
---- = --- -- = 1x = x
x x xx 1

- anonymous

so like that i can just "get rid of" the x part in the equation

- amistre64

not the whole thing... but you can get rid of alot of x s
how many x's do I have left in my solution up there?

- anonymous

the x^3 and the x^2 so 5 i think

- amistre64

x x x xx x
---- = --- -- = 1x = x
x x xx 1 ^
how many x's do
I have right here?

- anonymous

1

- amistre64

then we are left with 1 lonely little x that we have to leave in this equation.... right?

- anonymous

yea

- amistre64

you know already that we we multiply exponents we "add" then together.
When we do the opposite of multiplication (which is division) we do the opposite to the exponents too. we "SUBTRACT them.
x^3/x^2 = x^(3-2) = x^1 = x

- anonymous

ok so how do we plug this one lonely x into the equation

- amistre64

we just put it back were we got it from :) like this:
x y^2 z b^2
-------------
y a^5 b^2

- anonymous

ok what next. same thing to the b^2s

- amistre64

yep, that would be good.
b^2/b^2 is division.... so we SUBTRACT exponents.
b^(2-2) = b^0 = 1 which just disappears

- anonymous

ok so the b is just slapped onto the end of the top part of the equation without the exponent on it

- amistre64

there is no b to put back
if b^3 means bbb
and b^1 means b
b^0 means _______ no little b's to do anything with, zero b's, they are gone......

- anonymous

so on the actual equation after this last part the b will be no more

- amistre64

thats correct, they vanished......
x y^2 z
--------- now whats left to work with?
y a^5

- anonymous

the y

- amistre64

very good :)
so, can you show me how we work that out?

- anonymous

y^2/y^19(which is just nothing) subtract them and it leaves y

- amistre64

perfect!!

- amistre64

x y z
------ we appear to have come to the end :)
a^5

- anonymous

that 9 wasnt supposed to be there my finger sliped off the shift button when i hit the paraenthasis

- amistre64

i figured that :)

- anonymous

weird stinkin keyboards

- amistre64

i start getting so many typos it looks like im having a stroke......

- anonymous

back during the first of march i was in germany on a school junior senior trip and their keyboard was all kinds of weird

- amistre64

Yep different places have different setups for their own uses.
But this is the answer to our original problem....
x y z
------
a^5

- amistre64

I just want to go over one little detail.....

- anonymous

ok

- amistre64

when dividing with exponents, it is important to notice where the biggest one is. For example:
x^3/x^2 has the bigger number on top, so our answer goes back on top.
BUT
if the bottom has the bigger exponent, our answer will go back on the bottom.
for example:
x^2/x^7 has the bottom bigger than the top, we subtract like before but when were done our leftovers go on the bottom.
Does that make sense?

- anonymous

yea location of the bigger exponet is where the outcome of the division goes

- amistre64

very good :) Ciao

- anonymous

thank you so much if i have anymore questions ill just post them up and maybe be able to catch ya

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