Solve by Factoring x^-3=1/64

Subtract ${\displaystyle \frac{1}{64}}$ 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 ${\displaystyle 1^3}$.

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

Rewrite ${\displaystyle \frac{1}{64}}$ as ${\displaystyle {\left(\frac{1}{4}\right)}^3}$.

Since both terms are perfect cubes, factor using the difference of cubes formula, ${\displaystyle a^3-b^3=(a-b)(a^2+ab+b^2)}$ where ${\displaystyle a=\frac{1}{x}}$ and ${\displaystyle b=\frac{1}{4}}$.

To write ${\displaystyle \frac{1}{x}}$ as a fraction with a common denominator, multiply by ${\displaystyle \frac{4}{4}}$.

To write ${\displaystyle -\frac{1}{4}}$ as a fraction with a common denominator, multiply by ${\displaystyle \frac{x}{x}}$.

Combine the numerators over the common denominator.

To write ${\displaystyle \frac{1}{4x}}$ as a fraction with a common denominator, multiply by ${\displaystyle \frac{x}{x}}$.

To write ${\displaystyle \frac{1}{x^2}}$ as a fraction with a common denominator, multiply by ${\displaystyle \frac{4}{4}}$.

Combine the numerators over the common denominator.

To write ${\displaystyle \frac{x+4}{4x^2}}$ as a fraction with a common denominator, multiply by ${\displaystyle \frac{4}{4}}$.

To write ${\displaystyle \frac{1}{16}}$ as a fraction with a common denominator, multiply by ${\displaystyle \frac{x^2}{x^2}}$.

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 ${\displaystyle 0}$.