1. **State the problem:** Solve the equation $x^6 + x^6 = 2^{32}$ for $x$. 2. **Combine like terms:** Since both terms are $x^6$, we can add them: $$x^6 + x^6 = 2x^6$$ 3. **Rewrite the equation:** $$2x^6 = 2^{32}$$ 4. **Divide both sides by 2:** $$x^6 = \frac{2^{32}}{2} = 2^{31}$$ 5. **Solve for $x$ by taking the sixth root:** $$x = \pm \sqrt[6]{2^{31}} = \pm 2^{\frac{31}{6}}$$ 6. **Final answer:** $$x = \pm 2^{\frac{31}{6}}$$ This means $x$ can be either positive or negative $2^{31/6}$.