Simple but large fraction problem - no clue...

[Mercury3190]

Hard to explain, but in words: [(1 over a) + (1 over b)] that whole thing divided by: [(a^2) over (b^2)] with a minus 1 next to that...
It looks like:

1 + 1
-- --
a b
----------
a^2
---- -1
b^2

I need to express it in simplest form, and when I asked my teacher I was told the answer was:

b over [(a^2) - (ab)], or: b
--------
(a^2) - ab

Now I need to show the work but I don't understand this at all...
Please show me the steps to this process so I know from now on.

 

[stapel]

Recall that, to divide by a fraction, one inverts the "bottom" fraction and converts to multiplication. For instance:

. . . . .\(\displaystyle \large{\frac{\left(\frac{3}{5}\right)}{\left(\frac{4}{7}\right)}\,= \,\left(\frac{3}{5}\right)\,\left(\frac{7}{4}\right)\,= \,\frac{21}{20}}\)

Follow this same process with your exercise.

Eliz.

 

[Mercury3190]

Still stuck...

Thanks for the help there but I don't know how to apply it to my problem, with pluses and minuses and that -1...
Could you explain a bit more please? :?

 

[stapel]

To prepare for the inversion part, a good first step would be to combine the fractions in the top and bottom portions of the expression. So take the following:

. . . . .\(\displaystyle \large{\frac{1}{a}\,+\,\frac{1}{b}}\)

. . . . .\(\displaystyle \large{\frac{a^2}{b^2}\,-\,1}\)

...convert each to its common denominator ("ab" in the first case; "b²" in the second), and combine the terms into one fraction. You will then have "(fraction)/(fraction)", which you can invert and multiply.

If you get stuck, please reply showing your work. Thank you.

Eliz.

 

[Mercury3190]

Thanks!

Thank you everybody for helping me
That's a really neat computer program you have to be able to draw in equations like that - how do you do that (especially on this site) - or do you have to buy it?
Thanks again, I printed out those instructions to use in the future.

 

[soroban]

Hello, Mercury3190!

Simplify: \(\displaystyle \L\,\frac{\frac{1}{a}\,+\,\frac{1}{b}}{\frac{a^2}{b^2}\,-\,1}\)

When give a complex fraction (one with more than two "levels"),
\(\displaystyle \;\;\)multiply top and bottom by the LCD of the denominators. **


In this problem, the denominators are: \(\displaystyle \,a,\:b,\:b^2\) . . . the LCD is: \(\displaystyle \,ab^2\)

Multiply: \(\displaystyle \L\:\frac{ab^2\cdot\left(\frac{1}{a}\,+\,\frac{1}{b}\right)}{ab^2\cdot\left(\frac{a^2}{b^2}\,-\,1\right)}\;=\;\frac{\not{a}b^2\cdot\left(\frac{1}{\not{a}}\right)\,+\,ab^{\not{2}}\left(\frac{1}{\not{b}}\right)}{a\not{b^2}\left(\frac{a^2}{\not{b^2}}\right)\,-\,ab^2(1)} \;= \;\frac{b^2\,+\,ab}{a^3\,-\,ab^2}\)

Factor: \(\displaystyle \L\:\frac{b(b\,+\,a)}{a(a^2\,-\,b^2)}\:=\:\frac{b(\sout{a\,+\,b})}{a(a\,-\,b)(\sout{a\,+\,b})} \;= \;\frac{b}{a(a\,+\,b)}\)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

**\(\displaystyle \;\;\)Am I the only one that knows this?

 

[Denis]

Well, to each his own; my "own" with these (guess what N and D stand for!):

N = 1/a + 1/b = (a + b) / ab

D = a^2 / b^2 - 1 = (a^2 - b^2) / b^2

N * (1/D) = b^2 (a + b) / [ab(a^2 - b^2)] = b(a + b) / [a(a + b)(a - b)] = b / [a(a - b)]

 

[stapel]

Re: Thanks!

That's a really neat computer program you have to be able to draw in equations like that - how do you do that...

To learn how to use LaTeX, follow the lower links in the "Forum Help" pull-down menu at the very top of the page.

Note: You can review how others have formatted their posts by hitting "quote" or, at least in some browsers, putting your mouse over the formatted math portion.

Eliz.