# Math Help - Surds?

1. ## Surds?

Simplify

((108^1/2) + 10)^1/3 - ((108^1/2) - 10)^1/3

I gave it many tries but still got no headstart.

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 $x$ be this number. Cube both sides by using the fact $(a-b)^3 = a^3 - 3ab(a-b) - b^3$.
This gives us,
$x^3 = 20 - 3\sqrt[3]{8}x$
Thus,
$x^3 + 6x = 20$.
It is 'easy' to see that $x=2$ is a solution.

Thanks.

Let $a$ be the first radical and $b$ be the second radical. So you want to find the value of $a-b$. Let $x=a-b$ when you cube both sides you get, $x^3 = a^3 - 3ab(a-b) - b^2$. Thus, $x^3 = a^3 - b^3 - 3abx$. Now $a^3 - b^3 = 20$, that is easy to see because when you cube them the cube root goes away and the square roots cancel. And $ab$ is the difference of two square because $ab = \sqrt[3]{\sqrt{108}+10}\cdot \sqrt[3]{\sqrt{108}-10} = \sqrt[3]{108-100} = 2$.