1. **State the problem:** Simplify the expression $$\sqrt{x} - 20\sqrt{x^{10}} + 4\sqrt{x^2} - 6\sqrt{x^3} + 8\sqrt{x^4}$$ in radical form.\n\n2. **Rewrite each term using exponents:** Recall that $$\sqrt{x^n} = (x^n)^{\frac{1}{2}} = x^{\frac{n}{2}}.$$\n\nTherefore, the terms become:\n- $$\sqrt{x} = x^{\frac{1}{2}}$$\n- $$\sqrt{x^{10}} = x^{\frac{10}{2}} = x^5$$\n- $$\sqrt{x^2} = x^{\frac{2}{2}} = x^1 = x$$\n- $$\sqrt{x^3} = x^{\frac{3}{2}}$$\n- $$\sqrt{x^4} = x^{\frac{4}{2}} = x^2$$\n\n3. **Rewrite the expression with these powers:**\n$$x^{\frac{1}{2}} - 20x^5 + 4x - 6x^{\frac{3}{2}} + 8x^2$$\n\n4. **Group terms if possible:** Each term has a distinct power of $$x$$, so no like-term combination is possible. The simplified expression in exponent form is:\n$$x^{\frac{1}{2}} - 20x^5 + 4x - 6x^{\frac{3}{2}} + 8x^2$$\n\n5. **Convert back to radical form if necessary:**\n- $$x^{\frac{1}{2}} = \sqrt{x}$$\n- $$x^5 = (\sqrt{x})^{10}$$ (already simplified but keep as $$x^5$$ or $$\sqrt{x^{10}}$$ according to preference)\n- $$x = \sqrt{x^2}$$\n- $$x^{\frac{3}{2}} = x^{1} \cdot x^{\frac{1}{2}} = x \sqrt{x} = \sqrt{x^3}$$\n- $$x^2 = \sqrt{x^4}$$\n\nHence the expression in radical form is:\n$$\sqrt{x} - 20\sqrt{x^{10}} + 4\sqrt{x^2} - 6\sqrt{x^3} + 8\sqrt{x^4}$$\n\nThis matches the original expression, confirming the simplest radical form is the one given originally (no further simplification by combining like terms is possible).