1. **State the problem:** We are given the formula relating surface area $A$ and volume $V$ of a cube: $$A = 6V^{\frac{2}{3}}$$ and the surface area $A = 486$ square inches. We need to find the volume $V$ in cubic inches. 2. **Write down the formula:** $$A = 6V^{\frac{2}{3}}$$ 3. **Substitute the known value:** $$486 = 6V^{\frac{2}{3}}$$ 4. **Isolate $V^{\frac{2}{3}}$ by dividing both sides by 6:** $$\frac{486}{6} = \cancel{6}V^{\frac{2}{3}} \div \cancel{6}$$ $$81 = V^{\frac{2}{3}}$$ 5. **To solve for $V$, raise both sides to the power of $\frac{3}{2}$ (the reciprocal of $\frac{2}{3}$):** $$\left(81\right)^{\frac{3}{2}} = \left(V^{\frac{2}{3}}\right)^{\frac{3}{2}}$$ $$\left(81\right)^{\frac{3}{2}} = V$$ 6. **Calculate $\left(81\right)^{\frac{3}{2}}$:** First, $81 = 9^2$, so $$\left(9^2\right)^{\frac{3}{2}} = 9^{2 \times \frac{3}{2}} = 9^3 = 729$$ 7. **Final answer:** $$V = 729$$ cubic inches. **Explanation:** We used the given formula to express volume in terms of surface area, isolated the volume term, and then used exponent rules to solve for $V$. The volume of the cube is 729 cubic inches.