Solve by Factoring x^(-2/3)=9

Subtract ${\displaystyle 9}$ from both sides of the equation.

Rewrite the expression using the negative exponent rule ${\displaystyle b^{-n}=\frac{1}{b^n}}$.

Rewrite ${\displaystyle 1}$ as ${\displaystyle 1^2}$.

Rewrite ${\displaystyle x^{\frac{2}{3}}}$ as ${\displaystyle {\left(x^{\frac{1}{3}}\right)}^2}$.

Rewrite ${\displaystyle \frac{1^2}{{\left(x^{\frac{1}{3}}\right)}^2}}$ as ${\displaystyle {\left(\frac{1}{x^{\frac{1}{3}}}\right)}^2}$.

Rewrite ${\displaystyle 9}$ as ${\displaystyle 3^2}$.

Since both terms are perfect squares, factor using the difference of squares formula, ${\displaystyle a^2-b^2=(a+b)(a-b)}$ where ${\displaystyle a=\frac{1}{x^{\frac{1}{3}}}}$ and ${\displaystyle b=3}$.

To write ${\displaystyle 3}$ as a fraction with a common denominator, multiply by ${\displaystyle \frac{x^{\frac{1}{3}}}{x^{\frac{1}{3}}}}$.

Combine the numerators over the common denominator.

To write ${\displaystyle -3}$ as a fraction with a common denominator, multiply by ${\displaystyle \frac{x^{\frac{1}{3}}}{x^{\frac{1}{3}}}}$.

Combine ${\displaystyle -3}$ and ${\displaystyle \frac{x^{\frac{1}{3}}}{x^{\frac{1}{3}}}}$.

Combine the numerators over the common denominator.

If any individual factor on the left side of the equation is equal to ${\displaystyle 0}$, the entire expression will be equal to ${\displaystyle 0}$.