The numerator is equivalent to

The denominator is equivalent to

Dividing the numerator by the denominator, one gets

It's much easier to use factoring and canceling than it is to use long division for this problem. 9x² – 1 is a difference of squares. The difference of squares formula is a² – b² = (a + b)(a – b). So 9x² – 1 = (3x + 1)(3x – 1). Putting the numerator and denominator together, (9x² – 1) / (3x – 1) = (3x + 1)(3x – 1) / (3x – 1) = 3x + 1.

Factor both the numerator and the denominator which gives us the following:

After cancelling we get

There is a common factor in the numerator.   Pull out the common factor and rewrite the numerator.

Factorize the denominator.

Cancel the  term in the numerator and denominator.

The answer is: 

First factor the numerators and denominators of the two fractions. This allows us to re-write the original problem like this:

Now we can cancel terms that appear on both the top and the bottom, since they will divide to be a factor of . This means we can can cancel the top and bottom, the top and bottom , and the top and bottom . This leaves us with the following answer:

First, completely factor all 4 quadratics:

Now we can cancel all factors that appear on both the top and the bottom, because those will divide to a factor of . We quickly realize that all of the factors can be crossed off. This means that all of the factors have been divided to . This leves us with the following answer:

First, completely factor everything that can possibly be factored. This includes both numerators and the second denominator:

Now we can cancel everything that appears both on the top and the bottom, since it will divide to be a factor of :

We can simplify this by multiplying  and .

This leaves us with the following answer:

I would first start by simplifying the numerator by getting rid of the negative exponents: . Then, combine the denominator fractions into one fraction: . At this point, we're dividing fractions so we have to multiply by the reciprocal of the second fraction: . Multiply straight across to get: . Make sure it can't be simplified (it can't)!

First, combine the top two fractions. The common denominator between the two is Therefore, you just have to offset the first fraction so that it becomes . Then, combine the numerators to get . So at this point, we have: . This is essentially a dividing fractions problem. When we divide fractions, we have to make the second fraction its reciprocal (flip it!) and then multiply the two. . The 's cross out so your final answer is: .

When you divide by a fraction you must multiply by its reciprocal to get the correct quotient.

Factor where able:

Cancel like terms: