1. **State the problem:** We want to find the values of $x$ for which the expression $\sqrt[4]{56x^3}$ is defined. 2. **Recall the rule for even roots:** The fourth root (an even root) of a number is defined only if the number inside the root is non-negative (i.e., $\geq 0$). 3. **Set the radicand non-negative:** $$56x^3 \geq 0$$ 4. **Analyze the inequality:** Since 56 is positive, the sign depends on $x^3$. 5. **Sign of $x^3$:** - $x^3 \geq 0$ means $x \geq 0$. 6. **Conclusion:** The expression $\sqrt[4]{56x^3}$ is defined for all $x$ such that $$x \geq 0$$ **Final answer:** $x \geq 0$