Rationalise the denominator of the following fraction: 9/((root13)-1). Write your answer in its simplest form.

Firstly, a quick overview of rationalising the denominator. For this, we want to make sure that the bottom half of the fraction (the denominator) is rationalised (and so doesn't contain any surds any more). The denominator of this fraction is a binomial term (has two things added or minused together), and so we need to use the conjugate of this. The conjugate means flipping the sign in between the two terms. From then, we can use our standard procedure for rationalising the denominator - multiplying the fraction by the conjugate divided by the conjugate, as shown below:9/(root13-1)*(root13+1)/(root13+1)=(9root13+9)/(13-root13+root13-1)We can then simplify this fraction by cancelling out the root13 terms on the bottom, and dividing the whole thing by 3, to give:(9root13+9)/(13-root13+root13-1)=(9root13+9)/12=(3root13+3)/4Which is our simplified answer, as required in the question.