Question

How can I solve this?

Answer 1

D. `(xy-1)/(xy)`

Explanation:

`(1-x^-3y^-3)/(x^-2y^-2+x^-1y^-1+1)`

When you have negative exponents, you need to factor out the term with the smallest exponent.

In the numerator, it's `x^-3y^-3`. In the denominator it's `x^-2y^-2`.

`((x^-3y^-3)(x^3y^3-1))/((x^-2y^-2)(1+xy+x^2y^2))`

If you look at the second factor in the numerator, you can identify it as the cubic trinomial `a^3-b^3`.

`a^3-b^3=(a-b)(a^2+ab+b^2)`

Therefore, you can use this formula to simplify the second factor of the numerator.

`((x^-3y^-3)(xy-1)(x^2y^2+xy+1))/((x^-2y^-2)(1+xy+x^2y^2))`

Although they are written in different orders, the two right most factors in the numerator and denominator are the same. You can now cancel them out.

`((x^-3y^-3)(xy-1)cancel(x^2y^2+xy+1))/((x^-2y^-2)cancel(1+xy+x^2y^2))`

`((x^-3y^-3)(xy-1))/(x^-2y^-2)`

Last, you have one more factor that can be canceled out. You can do this by simplifying the first factor in the numerator.

`((x^-1y^-1)(x^-2y^-2)(xy-1))/(x^-2y^-2)`

`((x^-1y^-1)cancel(x^-2y^-2)(xy-1))/cancel(x^-2y^-2)`

Now you are left with this:

`(x^-1y^-1)(xy-1)`

You can rewrite the first term into a fraction and then simplify.

`1/(xy)(xy-1)`

`(xy-1)/(xy)`

If done correctly, you should get D as the right answer!