1. **State the problem:** Simplify the expression $$\left(\frac{(x^{12})^{0.25}(216x^{9})}{(3x)^{6}(x^{18})^{0.5}}\right)^{-1/3}$$ 2. **Recall exponent rules:** - Power of a power: $\left(a^m\right)^n = a^{mn}$ - Product of powers: $a^m \cdot a^n = a^{m+n}$ - Quotient of powers: $\frac{a^m}{a^n} = a^{m-n}$ - Negative exponent: $a^{-m} = \frac{1}{a^m}$ 3. **Simplify each part inside the fraction:** - $(x^{12})^{0.25} = x^{12 \times 0.25} = x^3$ - $216x^9$ stays as is - $(3x)^6 = 3^6 x^6 = 729 x^6$ - $(x^{18})^{0.5} = x^{18 \times 0.5} = x^9$ 4. **Rewrite the fraction:** $$\frac{x^3 \cdot 216 x^9}{729 x^6 \cdot x^9} = \frac{216 x^{3+9}}{729 x^{6+9}} = \frac{216 x^{12}}{729 x^{15}}$$ 5. **Simplify the coefficients and variables:** $$= \frac{\cancel{216}^\frac{}{}}{\cancel{729}^\frac{}{}} \cdot x^{12-15} = \frac{216}{729} x^{-3}$$ Simplify $\frac{216}{729}$ by dividing numerator and denominator by 27: $$\frac{216 \div 27}{729 \div 27} = \frac{8}{27}$$ So the fraction inside parentheses is: $$\frac{8}{27} x^{-3}$$ 6. **Apply the outer exponent $-\frac{1}{3}$:** $$\left(\frac{8}{27} x^{-3}\right)^{-\frac{1}{3}} = \left(\frac{8}{27}\right)^{-\frac{1}{3}} \cdot \left(x^{-3}\right)^{-\frac{1}{3}}$$ 7. **Simplify each part:** - $\left(\frac{8}{27}\right)^{-\frac{1}{3}} = \left(\frac{27}{8}\right)^{\frac{1}{3}} = \frac{27^{1/3}}{8^{1/3}} = \frac{3}{2}$ - $\left(x^{-3}\right)^{-\frac{1}{3}} = x^{-3 \times -\frac{1}{3}} = x^{1} = x$ 8. **Final simplified expression:** $$\frac{3}{2} x$$