Question

Can anyone help me divide a rational expression?

Find the Quotient and Simplify.

7x^7 over 3x^5 divided by 14x over 9x^3

Answer 1

if you divide by a fraction, you can "flip it" and multiply by it.
Can you do that for this problem ?

Answer 2

Yes, I know I have to flip the second fraction. I'm just not sure how to find the common denominator

Answer 3

you don't need a common denominator when multiplying (or dividing)
you should get
\[ \frac{7x^7}{3x^5} \cdot \frac{9x^3}{14x} \]
when multiplying you multiply top times top and bottom times bottom
but you can first divide (it's usually easier)

Answer 4

order can be changed when multiplying
for example, the top can be written as 7*9 * x^7 * x^3
and at the bottom 14*3*x^5*x
\[ \frac{ 7 \cdot 9 \cdot x^7 \cdot x^3}{14 \cdot 3 \cdot x^5 \cdot x}\]
and you can divide the numbers

Answer 5

you can "split apart" that fraction, and write it like this
\[ \frac{7}{14} \cdot \frac{9}{3} \cdot \frac{x^7 \cdot x^3}{x^5\cdot x} \]

Answer 6

people don't usually bother, but it might make it easier to see how to simplify the numbers
can you simplify 7/14 ?

Answer 7

yes that would be 1/2

Answer 8

For the x's you use the idea that
\[ \frac{x}{x} = 1\]
anything divided by itself is 1
so if you see x^7 (which is short for x*x*x*x*x*x*x)
and you see x^5 in the bottom: x*x*x*x*x
you can see 5 of the x's in the bottom can divide into 5 in the top (you get 1*1*1*1*1)
leaving x*x up top

yes 7/14 is ½
what about 9/3 ?

Answer 9

9/3 would be \[\frac{ 1 }{ 3 }\]

Answer 10

other way round , right ?

Answer 11

oops! yes, other way around

Answer 12

ok, make a note: the numbers simplify to ½ * 3 or 3/2

can you do the x's ?

Answer 13

I think so. Would the final answer be 3x^11 over 2^6?

Answer 14

how did you get that ?

Answer 15

ignore the numbers and look just at \[ \frac{x^7 x^3}{x^5x} \]

Answer 16

Don't you add the exponents?

Answer 17

with all those x's it's a pain to type out, so let's do a small problem
\[ \frac{x^3}{x} \]
as you know x^3 means x*x*x
\[ \frac{x\cdot x\cdot x}{x} \]
and as you know we can make that into separate fractions
\[ \frac{x}{x} \cdot \frac{x}{1}\cdot \frac{x}{1} \]
notice you multiply top times top and bottom times bottom we get what we started with

Answer 18

we can simply by know x/x is 1
and 1 times anything is the anything
in other words we get
\[ \frac{x}{1}\cdot \frac{x}{1} = \frac{x^2}{1}= x^2 \]

the fast way is to subtract the bottom exponent from the top exponent
3-1= 2 and write x^2 right away

Answer 19

yes, you can add exponent when multiplying
x^2 * x^2 means x*x * x*x or in short hand x^4
the fast way is to do 2+2 to get 4 for the exponent.

so you could do the top and get x^(7+3) = x^10
and the bottom you get x^(5+1) = x^6
and you can divide x^6 into x^10 to get x^4

Answer 20

Ok I'm getting a little lost now

Answer 21

try this one:
\[ \frac{x}{x} \]
what does the simplify to ?

Answer 22

1

Answer 23

and you might know if we put in an exponent of 1
x^1 = x
(we don't usually bother to show an exponent if it's one)
\[ \frac{x^1}{x^1} \]
if we subtract the exponents top - bottom : 1- 1=0 we get
\[ \frac{x^1}{x^1} =x^0 =1\]

Answer 24

I'm forgetting what my original problem is now

Answer 25

anything to the 0 power is 1.
(make a note of that... it may not be obvious)

Answer 26

we will do you problem in just a minute

can you simplify \[ \frac{x\cdot x}{x} \] ?

Answer 27

notice it's the same as simplifying
\[ \frac{x}{x} \cdot \frac{x}{1} \]

Answer 28

it would still be 1

Answer 29

or x^2 over 1x

Answer 30

x/x is 1
you can "replace" the x/x with 1, but you would then have
\[ \frac{x}{x} \cdot \frac{x}{1} =1 \cdot \frac{x}{1} \]

Answer 31

This is the idea you want to learn
\[ \frac{x\cdot x}{x} = \frac{x}{x} \cdot \frac{x}{1} = 1 \cdot \frac{x}{1} =\frac{x}{1}\]

Answer 32

I'm running out of time and just need the answer or at least simple steps to the answer

Answer 33

add the exponents in the top
what do you get ?

Answer 34

10

Answer 35

now do the same in the bottom

Answer 36

6

Answer 37

and because you are dividing do 10 - 6 to get the exponent for x

Answer 38

4

Answer 39

in other words x^4
with the numbers you get
\[ \frac{3x^4}{2} \]

Answer 40

and that is found by multiplying the fractions and then simplifying them? Thank you for all your help

Answer 41

You can also reduce each fraction before dividing them.

\(\Large \dfrac{7x^7}{3x^5} \div \dfrac{14x}{9x^3} \)

\(\Large =\dfrac{7x^2}{3} \div \dfrac{14}{9x^2} \)

\(\Large =\dfrac{7x^2}{3} \times \dfrac{9x^2}{14} \)

\(\Large =\dfrac{7 \times 3 \times 3 x^4}{3 \times 2 \times 7} \)

\(\Large =\dfrac{\cancel{7} \times \cancel{3} \times 3 x^4}{\cancel{3} \times 2 \times \cancel{7}} \)

\(\Large =\dfrac{3 x^4}{ 2}\)