Homework Help: Simplifying using index laws

1. Feb 26, 2013

1. The problem statement, all variables and given/known data
Simplify; expressing with positive indices.

$$\frac{x^{-2} - y^{-2}}{x^{-1} - y^{-1}}$$

3. The attempt at a solution
Hello, I'm doing first year uni math and over the holidays, I forgotten my index laws and as a result I'm stuck on this question :uhh:

I know that the answer is:

$$\frac{x+y}{xy}$$

However I cannot figure out which laws relate to the equation.

I don't think I can just cancel the x's and y's here, because of that takeaway sign. Also I cannot bring the denominator up because I don't know what the rule is for that.

Can someone please give me a hint towards the right direction?

Last edited by a moderator: Feb 26, 2013
2. Feb 26, 2013

Mentallic

This problem actually doesn't have too much to do with the index laws. Begin by removing the indices by converting them into fraction, then simplify from there.

OR if you prefer,
Notice that $x^{-2}-y^{-2}=(x^{-1})^2-(y^{-1})^2$ which is a difference of 2 squares.

Last edited by a moderator: Feb 26, 2013
3. Feb 26, 2013

Hello Mentallic, expanding the brackets gives me

$$\frac{(\frac{1}{x}+\frac{1}{y})(\frac{1}{x}-\frac{1}{y})}{\frac{1}{x}-\frac{1}{y}}$$

How do I proceed from here? because I cannot see a way which will give me postive index values. Do I take both brackets in the numerators down to the denominator?

Mod note: You don't need to use the HTML SIZE tags to make your LaTeX bigger - just use tex tags instead of itex tags.

Last edited by a moderator: Feb 26, 2013
4. Feb 26, 2013

Mentallic

Notice the common factor in the numerator and denominator?

5. Feb 28, 2013

Ahhh... I see it now.

Then I just multiply the remaining bracket by $\frac{xy}{xy}$ to finish off the question.

Thanks a lot Mentallic :)

6. Feb 28, 2013

HallsofIvy

My first thought was to multiply both numerator and denominator by $x^2y^2$:
$$\frac{x^{-2}- y^{-2}}{x^{-1}- y^{-1}}= \frac{y^2- x^2}{xy^2- x^2y}$$
$$= \frac{(x+ y)(x- y)}{xy(y- x)}$$

7. Mar 1, 2013

That works too! :)

If I have another problem I am stuck on relating to index laws, may I post it on this thread, or do I have to open a new topic?

8. Mar 1, 2013