How can I simplify this expression using index laws?

[miniradman]

Homework Statement

Simplify; expressing with positive indices.

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

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

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?

 

[Mentallic]

Homework Statement

Simplify; expressing with positive indices.

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

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

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?

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.

 

[miniradman]

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 positive 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.[/color]

 

[Mentallic]

Notice the common factor in the numerator and denominator?

 

[miniradman]

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 :)

 

[HallsofIvy]

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

 

[miniradman]

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?

 

[Mark44]

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?

For a new problem, please start a new thread.