# Canceling a Common Factor

Staff Emeritus
Science Advisor

## Homework Statement

I've got a fraction here:

$\frac{14+14\sqrt{3}}{-8}$

Why is it you can take a 2 out of the bottom and top to make it the following?

$\frac{7+7\sqrt{3}}{-4}$

I'm lost in figuring out how this works. I thought the top was like having (14+14x), where you can take a 14 out of each term and make it 14(1+x).

## Answers and Replies

Mentallic
Homework Helper
What does
$$\frac{14y}{2x}$$
equal to?

Also, you can move the negative sign from the denominator to the numerator by "taking out" a -1 from both the numerator and denominator.

SteamKing
Staff Emeritus
Science Advisor
Homework Helper
Really?

Code:
Lookit:
14 + 14*Sqrt(3)      2*7 + 2*7 * Sqrt(3)     2*[7 + 7 * Sqrt (3)]     7 + 7 * Sqrt (3)
---------------- =   ------------------- =   -------------------- = ----------------
-8                     2*(-4)                   2 * (-4)               -4

Staff Emeritus
Science Advisor
I understand that 14y/2x = 7y/x.

I thought you couldn't divide the original fraction that way because it's still adding up there and you had to take a factor out or something first.

SteamKing
Staff Emeritus
Science Advisor
Homework Helper
Multiplication distributes over addition, so a*(b + d) = a*b + a*d

Mentallic
Homework Helper
I understand that 14y/2x = 7y/x.

I thought you couldn't divide the original fraction that way because it's still adding up there and you had to take a factor out or something first.

Yes, that's true, but look at what you said earlier

I'm lost in figuring out how this works. I thought the top was like having (14+14x), where you can take a 14 out of each term and make it 14(1+x).

So in this case, the numerator is $14(1+\sqrt{3})$ so we can now let $y=1+\sqrt{3}$.

Staff Emeritus
Science Advisor
So in this case, the numerator is $14(1+\sqrt{3})$ so we can now let $y=1+\sqrt{3}$.

Arrghh... I had my answer as $\frac{7(1+\sqrt{3})}{4}$ , which was apparently wrong, whereas $\frac{7+7\sqrt{3})}{4}$ was correct.

Office_Shredder
Staff Emeritus
Science Advisor
Gold Member
Arrghh... I had my answer as $\frac{7(1+\sqrt{3})}{4}$ , which was apparently wrong, whereas $\frac{7+7\sqrt{3})}{4}$ was correct.

Those are both the exact same number, and are equally correct actually (well, except for the missing minus sign)

Staff Emeritus
Science Advisor
Those are both the exact same number, and are equally correct actually (well, except for the missing minus sign)

Ah yes, forgot the negative.
At least I got it figured out. I was so confused...