1. **State the problem:** Solve the equation $$\frac{x^{\frac{3}{2}}}{x-6} = x - 8$$ for $x$. 2. **Rewrite the equation:** Multiply both sides by $x-6$ (assuming $x \neq 6$) to eliminate the denominator: $$x^{\frac{3}{2}} = (x - 8)(x - 6)$$ 3. **Expand the right side:** $$(x - 8)(x - 6) = x^2 - 6x - 8x + 48 = x^2 - 14x + 48$$ 4. **Rewrite the equation:** $$x^{\frac{3}{2}} = x^2 - 14x + 48$$ 5. **Substitute:** Let $y = \sqrt{x}$, so $x = y^2$ and $x^{\frac{3}{2}} = y^3$. 6. **Rewrite in terms of $y$:** $$y^3 = (y^2)^2 - 14(y^2) + 48 = y^4 - 14y^2 + 48$$ 7. **Bring all terms to one side:** $$0 = y^4 - 14y^2 - y^3 + 48$$ 8. **Rearranged:** $$y^4 - y^3 - 14y^2 + 48 = 0$$ 9. **Try to find rational roots:** Test possible roots using Rational Root Theorem (factors of 48). 10. **Test $y=3$:** $$3^4 - 3^3 - 14 \times 3^2 + 48 = 81 - 27 - 126 + 48 = -24 \neq 0$$ 11. **Test $y=4$:** $$4^4 - 4^3 - 14 \times 4^2 + 48 = 256 - 64 - 224 + 48 = 16 \neq 0$$ 12. **Test $y=6$:** $$6^4 - 6^3 - 14 \times 6^2 + 48 = 1296 - 216 - 504 + 48 = 624 \neq 0$$ 13. **Test $y=2$:** $$2^4 - 2^3 - 14 \times 2^2 + 48 = 16 - 8 - 56 + 48 = 0$$ So, $y=2$ is a root. 14. **Divide polynomial by $(y-2)$:** Using polynomial division or synthetic division, quotient is: $$y^3 + y^2 - 12y - 24$$ 15. **Factor the cubic:** Try rational roots again. 16. **Test $y=2$ again:** $$2^3 + 2^2 - 12 \times 2 - 24 = 8 + 4 - 24 - 24 = -36 \neq 0$$ 17. **Test $y=-2$:** $$(-2)^3 + (-2)^2 - 12(-2) - 24 = -8 + 4 + 24 - 24 = -4 \neq 0$$ 18. **Test $y=3$:** $$27 + 9 - 36 - 24 = -24 \neq 0$$ 19. **Test $y=-3$:** $$-27 + 9 + 36 - 24 = -6 \neq 0$$ 20. **Use depressed cubic or numerical methods:** The cubic has no easy rational roots; approximate or use cubic formula. 21. **Approximate roots:** Using numerical methods, roots are approximately $y \approx 4.24$ and $y \approx -2.24$. 22. **Recall $y = \sqrt{x} \geq 0$, so discard negative roots.** 23. **Solutions for $y$:** $y = 2$ and $y \approx 4.24$. 24. **Convert back to $x$:** $$x = y^2$$ So, $$x = 2^2 = 4$$ $$x \approx (4.24)^2 = 17.98$$ 25. **Check for extraneous solutions:** - For $x=4$, denominator $x-6 = -2 \neq 0$, check original equation: $$\frac{4^{3/2}}{4-6} = \frac{8}{-2} = -4$$ Right side: $4 - 8 = -4$ matches. - For $x \approx 17.98$, denominator $17.98 - 6 = 11.98 \neq 0$, check original equation: $$\frac{(17.98)^{3/2}}{11.98} \approx \frac{(17.98)^{1.5}}{11.98} \approx \frac{76.2}{11.98} = 6.36$$ Right side: $17.98 - 8 = 9.98$ does not match, so discard. **Final solution:** $$\boxed{4}$$