MHB What are the Values of a, b, and c in the Real Root Equation 8x^3-3x^2-3x-1=0?

[anemone]

The real root of the equation $$8x^3-3x^2-3x-1=0$$ can be written in the form $$\frac{\sqrt[3]{a}+\sqrt[3]{b}+1}{c}$$, where $$a,\;b,\,c$$ are positive integers.

Find $$a+b+c$$.

 

[mathworker]

$$8x^3-3x^2-3x-1=0$$
$$9x^3=(x+1)^3$$
$$9=(1+\frac{1}{x})^3$$
$$x=\frac{1}{\sqrt[3]{9}-1}$$
by factorizing,
$$a=81,b=9,c=8$$
$$a+b+c=98$$