Express as a single fraction

Hi all,

This is my first post so thank you for reading it. I have started advanced level maths after 12 years of no studying of the subject. I have learned most of the stuff we have studied so far but it seems like I'm not alone in having issues with fractions. I've been asked to express the following as a single fraction:

| | __(m+n)^2__ | + | __M^2+mn__ |

| | m^2-n^2 | | n^2-mn |

I've tried many videos and pages online but to no avail. I'd be eternally grateful if someone could explain this step by step and that way I could practice it with other equations. I feel like I have missed how to do these with no numbers somewhere along the line.

Many thanks

Re: Express as a single fraction

Quote:

Originally Posted by

**lsmcal1984** Hi all,

This is my first post so thank you for reading it. I have started advanced level maths after 12 years of no studying of the subject. I have learned most of the stuff we have studied so far but it seems like I'm not alone in having issues with fractions. I've been asked to express the following as a single fraction:

| | __(m+n)^2__ | + | __M^2+mn__ |

| | m^2-n^2 | | n^2-mn |

I've tried many videos and pages online but to no avail. I'd be eternally grateful if someone could explain this step by step and that way I could practice it with other equations. I feel like I have missed how to do these with no numbers somewhere along the line.

Many thanks

1. I assume that you want to simplify

$\displaystyle \frac{(m+n)^2}{m^2-n^2} + \frac{m^2+mn}{n^2-mn}$

2. If so factor as much as possible:

$\displaystyle \frac{(m+n)^2}{m^2-n^2} + \frac{m^2+mn}{n^2-mn} = \frac{(m+n)(m+n)}{(m+n)(m-n)} + \frac{m(m+n)}{n(n-m)}$

3. For $\displaystyle m+n\ne 0$ you can cancel one factor in the 1st summand. Because $\displaystyle n-m=-(m-n)$ you'll get:

$\displaystyle \frac{(m+n)}{(m-n)} - \frac{m(m+n)}{n(m-n)}$

4. Obviously the common denominator is $\displaystyle n(m-n)$. Therefore:

$\displaystyle \frac{(m+n)}{(m-n)} - \frac{m(m+n)}{n(m-n)} = \frac{n(m+n)-m(m+n)}{n(m-n)}\\ = \frac{(n-m)(m+n)}{n(m-n)} = - \frac{m+n}n$

Re: Express as a single fraction

Thank you very much indeed - this puts together various things I've read on the Internet and I now know how to tackle algebraic fractions.

Much appreciated.

Leo