How do you simplify the expression (1/9 – 1/x) / (1/81 – 1/x²)?

To simplify the expression (1/9 - 1/x) / (1/81 - 1/x²), we will first rewrite the fractions with a common denominator.

Starting with the numerator:

• The common denominator between 9 and x is 9x. So we rewrite 1/9 as (x)/(9x) and 1/x as (9)/(9x).

This gives us:

1/9 - 1/x = (x - 9)/(9x)

Now, for the denominator:

• The common denominator between 81 and x² is 81x². We rewrite 1/81 as (x²)/(81x²) and 1/x² as (81)/(81x²).

This results in:

1/81 - 1/x² = (x² - 81)/(81x²)

Now our expression looks like this:

((x - 9)/(9x)) / ((x² - 81)/(81x²))

To divide by a fraction, we multiply by its reciprocal:

((x - 9)/(9x)) * ((81x²)/(x² - 81))

Then we can simplify:

• The x in the numerator and denominator will cancel out one x.
• The expression now is 81(x - 9)/(9(x² - 81)).
• Now notice that (x² - 81) can be factored using the difference of squares: (x - 9)(x + 9).

Now plug in this factorization:

81(x - 9)/(9((x - 9)(x + 9)))

We see that (x - 9) in the numerator and denominator cancel out:

So, we are left with:

81/(9(x + 9))

Simplifying this further gives:

9/(x + 9)

Thus, the simplified form of the original expression is:

9/(x + 9)