Question

simplify complex fractions:

Answer 1

complex as in with the imaginary "i"?

Answer 2

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

Answer 3

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

Answer 4

where the slash is its written like a fraction

Answer 5

So what is over what here?

Answer 6

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

Answer 7

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

Answer 8

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

Answer 9

the b^2.s cancel out

Answer 10

yes rational expressions

Answer 11

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

Answer 12

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

Answer 13

you mean the multiply part? or the flipping part?

Answer 14

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

Answer 15

make it easier on yourself and split them up...

a^4 means aaaa

c^2 mean cc

r^7 means rrrrrrr

Answer 16

now, x^3 times x^4 means

xxx xxxx = xxxxxxx = x^7

Answer 17

c^4 times c^5 means

cccc ccccc = ccccccccc = c^9

Answer 18

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

Answer 19

xxx yy z is all that is

Answer 20

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

Answer 21

what your confusing me

Answer 22

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

Answer 23

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

Answer 24

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

Answer 25

still not makin a whole lota sense

Answer 26

which part is not making sense?

Answer 27

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

Answer 28

Does:

xx y bb
------ = 1?
xx y bb

Answer 29

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

Answer 30

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.

Answer 31

lets keep it simpler, how would I start this?

x^2/y^3 // y/x^2

Answer 32

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

Answer 33

one step at a time, dont rush ahead... lets flip the bottom

x^2 x^2
---- * ---- now what do I do?
y^3 y

Answer 34

the x^2
----
y part

Answer 35

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

Answer 36

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

Answer 37

I already flipped it,

x^2 x^2
---- * ---- now what do I do?
y^3 y

Answer 38

i want to say crossmultiply

Answer 39

but thats usually not with letter

Answer 40

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

Answer 41

ok just add the exponets together

Answer 42

-------------->>
x^2 x^2 x^2 x^2
---- * ---- = ----------
y^3 y y^3 y
-------------->>

does this look right to you?

Answer 43

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

Answer 44

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

Answer 45

yea i think.

Answer 46

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

Answer 47

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

Answer 48

that looks more like it

Answer 49

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

Answer 50

ok

Answer 51

x^3 y^2 z
---------
a^2 b^2
---------------
a^3 x^2 y
--------
b^2

our first step is to do what?

Answer 52

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

Answer 53

ok....

x^3 y^2 z a^3 x^2 y
--------- / ----------
a^2 b^2 b^2

now what?

Answer 54

um flip the entire thing upside down or add the exponets

Answer 55

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

Answer 56

ok

Answer 57

just like this....

x^3 y^2 z b^2
--------- ----------
a^2 b^2 a^3 x^2 y

Answer 58

yea then add the exponets together

Answer 59

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.

Answer 60

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

Answer 61

line things up idk what your talking about

Answer 62

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

Answer 63

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

Answer 64

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

Answer 65

good we can do that...

x^3 y^2 z b^2
--------------------
x^2 y a^5 b^2

Answer 66

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

Answer 67

we cant "add" them together but watch this:

xx x
---- = what?
xx

Answer 68

5

Answer 69

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

Answer 70

i think

Answer 71

this is a fundamental concept in math.

whenever we "divide" a number by itself we get an answer of 1

1
---
8 | 8

right?

Answer 72

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

Answer 73

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......

Answer 74

when we divide a number by itself we get 1

9 divided by 9 = 9/9 = 1

Answer 75

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

Answer 76

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

Answer 77

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

Answer 78

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?

Answer 79

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

Answer 80

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

Answer 81

1

Answer 82

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

Answer 83

yea

Answer 84

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

Answer 85

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

Answer 86

we just put it back were we got it from :) like this:

x y^2 z b^2
-------------
y a^5 b^2

Answer 87

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

Answer 88

yep, that would be good.

b^2/b^2 is division.... so we SUBTRACT exponents.

b^(2-2) = b^0 = 1 which just disappears

Answer 89

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

Answer 90

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......

Answer 91

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

Answer 92

thats correct, they vanished......

x y^2 z
--------- now whats left to work with?
y a^5

Answer 93

the y

Answer 94

very good :)

so, can you show me how we work that out?

Answer 95

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

Answer 96

perfect!!

Answer 97

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

Answer 98

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