1. **State the problem:** Solve the equation $$c = \left( x \sqrt{3} \right)^{\frac{1}{3}} = \left(1 - \sqrt{2 - 5x} \right)^{\frac{1}{2}}$$ for $x$. 2. **Rewrite the equation:** We have $$\left( x \sqrt{3} \right)^{\frac{1}{3}} = \left(1 - \sqrt{2 - 5x} \right)^{\frac{1}{2}}$$ 3. **Square both sides** to eliminate the square root on the right: $$\left[ \left( x \sqrt{3} \right)^{\frac{1}{3}} \right]^2 = 1 - \sqrt{2 - 5x}$$ which simplifies to $$\left( x \sqrt{3} \right)^{\frac{2}{3}} = 1 - \sqrt{2 - 5x}$$ 4. **Isolate the square root term:** $$\sqrt{2 - 5x} = 1 - \left( x \sqrt{3} \right)^{\frac{2}{3}}$$ 5. **Square both sides again** to remove the square root: $$2 - 5x = \left[ 1 - \left( x \sqrt{3} \right)^{\frac{2}{3}} \right]^2$$ 6. **Expand the right side:** $$2 - 5x = 1 - 2 \left( x \sqrt{3} \right)^{\frac{2}{3}} + \left( x \sqrt{3} \right)^{\frac{4}{3}}$$ 7. **Bring all terms to one side:** $$0 = 1 - 2 \left( x \sqrt{3} \right)^{\frac{2}{3}} + \left( x \sqrt{3} \right)^{\frac{4}{3}} + 5x - 2$$ which simplifies to $$0 = -1 - 2 \left( x \sqrt{3} \right)^{\frac{2}{3}} + \left( x \sqrt{3} \right)^{\frac{4}{3}} + 5x$$ 8. **Let** $$y = \left( x \sqrt{3} \right)^{\frac{2}{3}}$$ so that $$y^2 = \left( x \sqrt{3} \right)^{\frac{4}{3}}$$. Rewrite the equation as $$0 = -1 - 2y + y^2 + 5x$$ 9. **Express $x$ in terms of $y$:** $$5x = 1 + 2y - y^2$$ $$x = \frac{1 + 2y - y^2}{5}$$ 10. **Recall the substitution:** $$y = \left( x \sqrt{3} \right)^{\frac{2}{3}} = \left( x^{\frac{1}{3}} \cdot 3^{\frac{1}{6}} \right)^2 = x^{\frac{2}{3}} \cdot 3^{\frac{1}{3}}$$ So $$y = 3^{\frac{1}{3}} x^{\frac{2}{3}}$$ 11. **Rewrite $x$ in terms of $y$ and substitute:** $$x = \left( \frac{y}{3^{\frac{1}{3}}} \right)^{\frac{3}{2}} = y^{\frac{3}{2}} \cdot 3^{-\frac{1}{2}}$$ 12. **Equate the two expressions for $x$:** $$y^{\frac{3}{2}} \cdot 3^{-\frac{1}{2}} = \frac{1 + 2y - y^2}{5}$$ 13. **Multiply both sides by 5 and $3^{\frac{1}{2}}$ to clear denominators:** $$5 y^{\frac{3}{2}} = 3^{\frac{1}{2}} (1 + 2y - y^2)$$ 14. **This is a transcendental equation in $y$ and can be solved numerically.** 15. **Once $y$ is found, compute $x$ using:** $$x = \left( \frac{y}{3^{\frac{1}{3}}} \right)^{\frac{3}{2}}$$ **Final answer:** The solution $x$ satisfies the equation above and can be found numerically. --- **Additional note:** The second equation $3xz=11$ is unrelated to the first and is ignored as per instructions.