Question 719718
First of all, please<ul><li>Include the instructions for the problem. Don't assume that there is only one obvious thing to do with an expression.</li><li>Put multiple-term numerators and denominators in parentheses. What you posted meant:
{{{1/root(3, 4) + root(3, 5)}}}
But I'm pretty sure you intended:
{{{1/(root(3, 4) + root(3, 5))}}}</li></ul>Assuming that the expression is:
{{{1/(root(3, 4) + root(3, 5))}}}
and the instructions are "Rationalize the denominator" (or something to that effect), then we will take advantage of the factoring pattern:
{{{(a+b)(a^2-ab+b^2) = a^3+b^3}}}
This pattern shows how a two-term expression times a certain three-term expression results in an expression of perfect cubes. The denominator we have to rationalize is a two-term expression which matches the pattern of (a+b). So the pattern shows how to turn that into an expression of perfect cubes. Since the cube of a cube root is rational, this is our path to a solution. In our denominator, the "a" is {{{root(3,4)}}} and the "b" is {{{root(3,5)}}}. If we multiply the numerator and denominator by the corresponding {{{a^2-ab+b^2}}} we will reach our goal:
{{{(1/(root(3, 4) + root(3, 5)))(((root(3,4))^2-(root(3,4))(root(3, 5))+(root(3, 5))^2)/((root(3,4))^2-(root(3,4))(root(3, 5))+(root(3, 5))^2))}}}
In the denominator the pattern tells us how it works out:
{{{((root(3, 4))^2-(root(3, 4))(root(3, 5))+(root(3, 5))^2)/((root(3, 4))^3 + (root(3, 5))^3)}}}
which simplifies to:
{{{(root(3,16)-root(3,20)+root(3, 25))/(4+5)}}}
then
{{{(root(3,16)-root(3,20)+root(3, 25))/9}}}