Simplify 1/( cube root of 9- cube root of 3)

$\frac{1}{\sqrt[3]{9} - \sqrt[3]{3}}$

Multiply $\frac{1}{\sqrt[3]{9} - \sqrt[3]{3}}$ by $\frac{{\sqrt[3]{9}}^{2} - \sqrt[3]{9} (- \sqrt[3]{3}) + {(- \sqrt[3]{3})}^{2}}{{\sqrt[3]{9}}^{2} - \sqrt[3]{9} (- \sqrt[3]{3}) + {(- \sqrt[3]{3})}^{2}}$ .

$\frac{1}{\sqrt[3]{9} - \sqrt[3]{3}} \cdot \frac{{\sqrt[3]{9}}^{2} - \sqrt[3]{9} (- \sqrt[3]{3}) + {(- \sqrt[3]{3})}^{2}}{{\sqrt[3]{9}}^{2} - \sqrt[3]{9} (- \sqrt[3]{3}) + {(- \sqrt[3]{3})}^{2}}$

Combine fractions.

$\frac{{\sqrt[3]{9}}^{2} - \sqrt[3]{9} (- \sqrt[3]{3}) + {(- \sqrt[3]{3})}^{2}}{(\sqrt[3]{9} - \sqrt[3]{3}) ({\sqrt[3]{9}}^{2} - \sqrt[3]{9} (- \sqrt[3]{3}) + {(- \sqrt[3]{3})}^{2})}$

Expand the denominator using the FOIL method.

$\frac{{\sqrt[3]{9}}^{2} - \sqrt[3]{9} (- \sqrt[3]{3}) + {(- \sqrt[3]{3})}^{2}}{{\sqrt[3]{9}}^{3} + \sqrt[3]{243} + \sqrt[3]{9} {(- \sqrt[3]{3})}^{2} - \sqrt[3]{3} {\sqrt[3]{9}}^{2} - \sqrt[3]{81} - \sqrt[3]{3} {(- \sqrt[3]{3})}^{2}}$

Simplify.

$\frac{{\sqrt[3]{9}}^{2} - \sqrt[3]{9} (- \sqrt[3]{3}) + {(- \sqrt[3]{3})}^{2}}{6}$

Simplify the numerator.

Rewrite ${\sqrt[3]{9}}^{2}$ as ${(9^{2})}^{\frac{1}{3}}$ .

$\frac{\sqrt[3]{9^{2}} - \sqrt[3]{9} (- \sqrt[3]{3}) + {(- \sqrt[3]{3})}^{2}}{6}$

Raise $9$ to the power of $2$ .

$\frac{\sqrt[3]{81} - \sqrt[3]{9} (- \sqrt[3]{3}) + {(- \sqrt[3]{3})}^{2}}{6}$

Rewrite $81$ as $3^{3} \cdot 3$ .

Factor $27$ out of $81$ .

$\frac{\sqrt[3]{27 (3)} - \sqrt[3]{9} (- \sqrt[3]{3}) + {(- \sqrt[3]{3})}^{2}}{6}$

Rewrite $27$ as $3^{3}$ .

$\frac{\sqrt[3]{3^{3} \cdot 3} - \sqrt[3]{9} (- \sqrt[3]{3}) + {(- \sqrt[3]{3})}^{2}}{6}$

Pull terms out from under the radical.

$\frac{3 \sqrt[3]{3} - \sqrt[3]{9} (- \sqrt[3]{3}) + {(- \sqrt[3]{3})}^{2}}{6}$

Multiply $- \sqrt[3]{9} (- \sqrt[3]{3})$ .

Multiply $- 1$ by $- 1$ .

$\frac{3 \sqrt[3]{3} + 1 \sqrt[3]{9} \sqrt[3]{3} + {(- \sqrt[3]{3})}^{2}}{6}$

Multiply $\sqrt[3]{9}$ by $1$ .

$\frac{3 \sqrt[3]{3} + \sqrt[3]{9} \sqrt[3]{3} + {(- \sqrt[3]{3})}^{2}}{6}$

Combine using the product rule for radicals.

$\frac{3 \sqrt[3]{3} + \sqrt[3]{9 \cdot 3} + {(- \sqrt[3]{3})}^{2}}{6}$

Multiply $9$ by $3$ .

$\frac{3 \sqrt[3]{3} + \sqrt[3]{27} + {(- \sqrt[3]{3})}^{2}}{6}$

Rewrite $27$ as $3^{3}$ .

$\frac{3 \sqrt[3]{3} + \sqrt[3]{3^{3}} + {(- \sqrt[3]{3})}^{2}}{6}$

Pull terms out from under the radical, assuming real numbers.

$\frac{3 \sqrt[3]{3} + 3 + {(- \sqrt[3]{3})}^{2}}{6}$

Apply the product rule to $- \sqrt[3]{3}$ .

$\frac{3 \sqrt[3]{3} + 3 + {(- 1)}^{2} {\sqrt[3]{3}}^{2}}{6}$

Raise $- 1$ to the power of $2$ .

$\frac{3 \sqrt[3]{3} + 3 + 1 {\sqrt[3]{3}}^{2}}{6}$

Multiply ${\sqrt[3]{3}}^{2}$ by $1$ .

$\frac{3 \sqrt[3]{3} + 3 + {\sqrt[3]{3}}^{2}}{6}$

Rewrite ${\sqrt[3]{3}}^{2}$ as ${(3^{2})}^{\frac{1}{3}}$ .

$\frac{3 \sqrt[3]{3} + 3 + \sqrt[3]{3^{2}}}{6}$

Raise $3$ to the power of $2$ .

$\frac{3 \sqrt[3]{3} + 3 + \sqrt[3]{9}}{6}$

The result can be shown in multiple forms.

Exact Form:

$\frac{3 \sqrt[3]{3} + 3 + \sqrt[3]{9}}{6}$

Decimal Form:

$1.56780542 \dots$

Simplify 1/( cube root of 9- cube root of 3)

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