# How do you rationalize the denominator and simplify 1/(1+root3x+root3(x^2)#?

Refer to explanation

#### Explanation:

Set $t = \sqrt[3]{x}$ hence we have that

$\frac{1}{1 + t + {t}^{2}}$

We know that ${t}^{3} - 1 = \left(t - 1\right) \cdot \left(1 + t + {t}^{2}\right) \implies 1 + t + {t}^{2} = \frac{{t}^{3} - 1}{t - 1}$

hence

$\frac{1}{1 + t + {t}^{2}} = \frac{1}{\frac{{t}^{3} - 1}{t - 1}} = \frac{t - 1}{{t}^{3} - 1}$

because $t = \sqrt[3]{x}$

we replace it and get

$\frac{\sqrt[3]{x} - 1}{\left({\left(\sqrt[3]{x}\right)}^{3}\right) - 1} = \frac{\sqrt[3]{x} - 1}{x - 1}$