Eliminate the irrationality in the denominator of the fraction, using polynomials

Question

Eliminate the irrationality in the denominator of the fraction, using polynomials:

$$\frac{1}{2+10^{1/3}-100^{1/3}}$$

Answer 1

The given number is equal to $$-\frac{14+8\cdot 10^{1/3}+3\cdot 100^{1/3}}{22}$$

Denote $x:=10^{1/3}$

Then, we have $$\frac{1}{2+x-x^2}=-\frac{1}{(x-2)(x+1)}=-\frac{1}{3}\cdot \frac{1}{x-2}+\frac{1}{3}\cdot \frac{1}{x+1}$$ $$=-\frac{1}{3}\cdot \frac{x^2+2x+4}{x^3-8}+\frac{1}{3}\cdot \frac{x^2-x+1}{x^3+1}$$

Now plug $x$ into the last expression to get the final result.