Solve 1/sqrt[3]{3-sqrt[3]{9}}=a+bsqrt[3]{3}+csqrt[3]{9}

$$\frac { 1 } { \sqrt[ 3 ] { 3 - \sqrt[ 3 ] { 9 } } } = a + b \sqrt[ 3 ] { 3 } + c \sqrt[ 3 ] { 9 }$$

Solve for b

$$b=-\frac{3^{\frac{2}{3}}a}{3}-\sqrt[3]{3}c+\frac{3^{\frac{4}{9}}}{3\sqrt[9]{3^{\frac{4}{3}}+2-3^{\frac{5}{3}}}}$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$a+b\sqrt[3]{3}+c\sqrt[3]{9}=\frac{1}{\sqrt[3]{3-\sqrt[3]{9}}}$$

Subtract a from both sides.

$$b\sqrt[3]{3}+c\sqrt[3]{9}=\frac{1}{\sqrt[3]{3-\sqrt[3]{9}}}-a$$

Subtract c\sqrt[3]{9} from both sides.

$$b\sqrt[3]{3}=\frac{1}{\sqrt[3]{3-\sqrt[3]{9}}}-a-c\sqrt[3]{9}$$

Reorder the terms.

$$\sqrt[3]{3}b=-\sqrt[3]{9}c-a+\frac{1}{\sqrt[3]{-\sqrt[3]{9}+3}}$$

The equation is in standard form.

$$\sqrt[3]{3}b=-\sqrt[3]{9}c-a+\frac{1}{\sqrt[3]{3-\sqrt[3]{9}}}$$

Divide both sides by \sqrt[3]{3}.

$$\frac{\sqrt[3]{3}b}{\sqrt[3]{3}}=\frac{-3^{\frac{2}{3}}c-a+\frac{1}{\sqrt[3]{3-3^{\frac{2}{3}}}}}{\sqrt[3]{3}}$$

Dividing by \sqrt[3]{3} undoes the multiplication by \sqrt[3]{3}.

$$b=\frac{-3^{\frac{2}{3}}c-a+\frac{1}{\sqrt[3]{3-3^{\frac{2}{3}}}}}{\sqrt[3]{3}}$$

Divide -a-3^{\frac{2}{3}}c+\frac{1}{\sqrt[3]{-3^{\frac{2}{3}}+3}} by \sqrt[3]{3}.

$$b=-\sqrt[3]{3}c-\frac{a}{\sqrt[3]{3}}+\frac{1}{3^{\frac{5}{9}}\sqrt[3]{\sqrt[3]{3}-1}}$$

Solve for a

$$a=-\sqrt[3]{3}b-3^{\frac{2}{3}}c+\frac{3^{\frac{7}{9}}}{3\sqrt[9]{3^{\frac{4}{3}}+2-3^{\frac{5}{3}}}}$$