 A compound fraction is a fraction whose numerator or denominator includes fractions. We can simplify these in several ways, but the simplest relies on three basic principles. You can always multiply an expression by one. One is any non-zero expression divided by itself, and multiplying a fraction by its denominator will eliminate the denominator. For example, if we want to simplify 8 divided by 1 plus two-thirds, notice the denominator includes a denominator of 3, so if we multiply the fraction by 3, we can eliminate the denominator. But we can't multiply part of the denominator by 3, we have to multiply the entire denominator by 3, and since that would change the value, we'll also multiply the numerator by 3. And in particular, this means we're multiplying the expression by three-thirds, or one. So we can rewrite by multiplying numerators and multiplying denominators, and we'll compute. 8 times 3 is 24, so that'll be our numerator, and at the denominator we have 1 plus two-thirds times 3, which will be… we can do the same thing if we have a compound rational expression. And it's helpful to keep in mind factored form is best. So let's simplify 8 over 1 plus 2 over x. So notice that the denominator includes a denominator of x, so we can begin by multiplying numerator and denominator by x. And let's go ahead and rewrite that still in factored form, because factored form is best, but we still have a fraction in the denominator, and we'd like to simplify it. Since we want to eliminate the compound fraction, we should expand the denominator, and we find, or we could take a more complicated compound fraction, and so it's important to remember it's OK to take small steps. So notice that the numerator includes a denominator of x plus 2, so we can begin by multiplying numerator and denominator by x plus 2. And remember, factored form is best will leave this denominator in its factored form, although since the numerator still has a bunch of fractions in it, we'll probably want to simplify it. Again, since we want to eliminate the fractions in the numerator, we'll expand and find. Now our numerator still includes a denominator of x, so we'll multiply numerator and denominator by x. Factored form is best, so we'll leave the denominator in the form of 2 times x times x plus 2, and since we want to eliminate the fractions in the numerator, we'll expand the numerator to get. And here we do have an option of removing the common factor to get our final answer as.