How Can Complex Fractions be Simplified?

[velox_xox]

First of all, thank you, everyone, for all your help with my math conundrums! I've really appreciated the patience and helpful hints - because I do want to learn it myself, so this method is just enough for me to cut the math problems down to size. I'm thankful. And, I have noticed my scores improved because I'm recognizing my trouble areas more often. So, to anyone out there shy to ask for help, I do suggest this forum. It's been very useful!

Homework Statement

Simplify. (a - b)/(a^-1 - b^-1)

Answer: -ab

This lesson is teaching that you can either simplify the numerator and denominator separately and divide; or multiply the whole thing by the LCD and then simpify.

Homework Equations

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The Attempt at a Solution

(a - b)/(a^-1 - b^-1)
Switched to a different format.
(a-b)/1 ÷ (a^-1 - b^-1)/1
Get rid of negative exponents.
(a-b)/1 ÷ 1/(a - b)
Invert the second fraction & switch to multiplication.
(a-b)/1 X (a-b)/1
Perform multiplication.
a² - 2ab - b²

 

[scurty]

(a-b)/1 ÷ (a^-1 - b^-1)/1
Get rid of negative exponents.
(a-b)/1 ÷ 1/(a - b)

Hi velox! This step is wrong! It might be easier to see the two cases you are confusing in fraction form: $a^{-1}-b^{-1}= \displaystyle \frac{1}{a}-\frac{1}{b}$ and $(a-b)^{-1} = \displaystyle \frac{1}{a-b}$

Whatever is being raised to the negative one gets flipped. All nearby operators stay the same. So in the first case that I showed above, the a and the b, individually, are raised to the negative one. So a and b, individually, get "flipped" and the minus sign stays where it is. In the second case the whole expression is raised to the negative one; so the whole expression gets flipped.

I hope that helps! Try getting the correct answer now!

 

[velox_xox]

Hi scurty, it's nice to have your help again!

That's right, the denominators need to be the same in subtraction with fractions, so I see that the separate 1/a - 1/b and 1/a-b aren't the same. I guess I need to watch out for that.

So, then...
(a - b)/(1/a - 1/b)
Use the LCD of 'ab' to create a common denominator.
{(a - b)/(b - a)/(ab)}
Turn the complex fraction into two simple ones.
(a - b)/1 ÷ (b - a)/(ab)
Invert the second fraction and change the sign to multiplication (and also change the denominator of the second fraction to the proper form).
(a - b)/1 X (ab)/(-a + b)
Factor out the '-1' from the denominator of the second fraction.
(a - b)/1 X (ab)/-1(a - b)
Simplify.
(ab)/-1
Divide.
-ab

Is this correct??

 

[scurty]

Absolutely! I see no problem with your method!

 

[velox_xox]

Well, that was fast. Thank you so much scurty! Have a good weekend!