Please help me with the following.....
Simplify
((108^1/2) + 10)^1/3 - ((108^1/2) - 10)^1/3
I gave it many tries but still got no headstart.
Very you solutions cubics algebraically?
Because let $\displaystyle x$ be this number. Cube both sides by using the fact $\displaystyle (a-b)^3 = a^3 - 3ab(a-b) - b^3$.
This gives us,
$\displaystyle x^3 = 20 - 3\sqrt[3]{8}x$
Thus,
$\displaystyle x^3 + 6x = 20$.
It is 'easy' to see that $\displaystyle x=2$ is a solution.
Let $\displaystyle a$ be the first radical and $\displaystyle b$ be the second radical. So you want to find the value of $\displaystyle a-b$. Let $\displaystyle x=a-b$ when you cube both sides you get, $\displaystyle x^3 = a^3 - 3ab(a-b) - b^2$. Thus, $\displaystyle x^3 = a^3 - b^3 - 3abx$. Now $\displaystyle a^3 - b^3 = 20$, that is easy to see because when you cube them the cube root goes away and the square roots cancel. And $\displaystyle ab$ is the difference of two square because $\displaystyle ab = \sqrt[3]{\sqrt{108}+10}\cdot \sqrt[3]{\sqrt{108}-10} = \sqrt[3]{108-100} = 2$.