### Instruction: Solve for $x$: $$\sqrt[3]{3-\frac{1}{x}}=-4$$.### Response: To solve for $x$, I want to isolate it on one side of the equation.
I can start by getting rid of the cube root by cubing both sides: $$(\sqrt[3]{3-\frac{1}{x}})^3=(-4)^3$$
Simplifying the cubes, I get: $$3-\frac{1}{x}=-64$$
Now I can add $\frac{1}{x}$ to both sides: $$3-\frac{1}{x}+\frac{1}{x}=-64+\frac{1}{x}$$
Simplifying, I get: $$3=-64+\frac{1}{x}$$
Next, I can subtract 3 from both sides: $$3-3=-64-3+\frac{1}{x}$$
Simplifying, I get: $$0=-67+\frac{1}{x}$$
I can rearrange this equation by adding 67 to both sides: $$0+67=-67+67+\frac{1}{x}$$
Simplifying, I get: $$67=\frac{1}{x}$$
Now I can solve for $x$ by taking the reciprocal of both sides: $$\frac{1}{67}=\frac{1}{\frac{1}{x}}$$