It is given that this is a rational number. Find $$\sqrt[3]{6\sqrt{3}+10}-\sqrt[3]{6\sqrt{3}-10}.$$

Solution:

Let, $$\sqrt[3]{6\sqrt{3}+10}-\sqrt[3]{6\sqrt{3}-10}=x$$. Then we are getting, $$\sqrt[3]{6\sqrt{3}+10}-\sqrt[3]{6\sqrt{3}-10}-x=0$$.

Now, we know that if $$a+b+c=0$$ then, $$a^3+b^3+c^3=3abc$$. Let, $$a=\sqrt[3]{6\sqrt{3}+10},b=-\sqrt[3]{6\sqrt{3}-10},c=-x$$

Then, we will get the degree 3 polynomial $$x^3+6x-20=0$$. As, $x$ is rational so we need the rational roots of this polynomial and as it is monic polynomial, so, rational root will have denominator 1(i.e; integer). Now, the roots will divide 20. Checking just few, x=2 satisfies the equation. Hence, $$x=2$$. The other roots will not satisfy as, if other roots are negative then it can't be a and if positive then it will be far away from the number we are getting from approximation of the given surd.