Question 957853
First of all, please put multiple-term numerators and denominators in parentheses. What you posted meant:
{{{1/root(3, 4) + root(3, 2) + 1}}}
But I'm quite sure you meant:
{{{1/(root(3, 4) + root(3, 2) + 1)}}}
which should be posted as:
1/(cube root 4+ cube root 2 +1)
Tutors are more likely to respond if problems are posted clearly.<br>
Clearing cube roots from a multiple-term denominator involves use of one or both of the following patterns:<ul><li>{{{a^3+b^3 = (a+b)(a^2-ab+b^2)}}}</li><li>{{{a^3-b^3 = (a-b)(a^2+ab+b^2)}}}</li></ul>A relatively fast way to simplify this requires that we recognizing that {{{root(3, 4) = (root(3, 2))^2}}}. Substituting this into our expression we get:
{{{1/((root(3, 2))^2 + root(3, 2) + 1)}}}
Now we need to recognize that this denominator matches the pattern of the second factor of {{{a^3-b^3 = (a-b)(a^2+ab+b^2)}}} with an "a" of {{{root(3, 2)}}} and a "b" of 1. The pattern shows us that if we multiply that factor by (a-b) then we get {{{a^3-b^3}}}. With all terms being perfect cubes, this will eliminate the cube roots.<br>
So we multiply the numerator and denominator by (a-b) with an "a" of {{{root(3, 2)}}} and a "b" of 1:
{{{(1/((root(3, 2))^2 + root(3, 2) + 1))((root(3, 2) - 1)/(root(3, 2) - 1))}}}
In the denominator, the pattern tells us what we get. In the numerator we just use the Distributive Property:
{{{(root(3, 2) - 1)/((root(3, 2))^3 - (1)^3))}}}
Simplifying...
{{{(root(3, 2) - 1)/(2-1)}}}
{{{(root(3, 2) - 1)/1}}}
{{{root(3, 2) - 1}}}