Question 248997
{{{root(3, 144) + root(3, 2/3) - 5root(3, 18)}}}
We cannot add or subtract any of these terms because they are not like terms (yet). But we can simplify them. There are two parts to simplifying radicals:<ul><li>Factoring out perfect powers of the root. (In this case we will be factoring out perfect cubes where possible.)</li><li>Rationalizing denominators.</li></ul>
The first cube root has a perfect square factor. The middle cube root needs to be rationalized:
{{{root(3, 8*18) + root(3, (2/3)(9/9)) - 5root(3, 18)}}}
{{{root(3, 8)*root(3, 18) + root(3, 18/27) - 5root(3, 18)}}}
{{{2*root(3, 18) + root(3, 18)/root(3, 27) - 5root(3, 18)}}}
{{{2*root(3, 18) + root(3, 18)/3 - 5root(3, 18)}}}
Each cube root is now simplified. And they are now like terms! So we can add and subtract them. Of course we need common denominators first:
{{{6*root(3, 18)/3 + root(3, 18)/3 - 15root(3, 18)/3}}}
{{{-8*root(3, 18)/3}}}
or
{{{((-8)/3)*root(3, 18)}}}