1. **State the problem:** Simplify and evaluate the expression $$\left( \frac{12^{\frac{1}{5}}}{27^{\frac{1}{5}}} \right)^{\frac{5}{2}}$$. 2. **Recall the exponent rules:** - When you have a fraction raised to a power, apply the power to numerator and denominator separately: $$\left( \frac{a}{b} \right)^m = \frac{a^m}{b^m}$$. - When you have a power raised to another power, multiply the exponents: $$\left(a^m\right)^n = a^{mn}$$. 3. **Apply the power to the fraction:** $$\left( \frac{12^{\frac{1}{5}}}{27^{\frac{1}{5}}} \right)^{\frac{5}{2}} = \frac{\left(12^{\frac{1}{5}}\right)^{\frac{5}{2}}}{\left(27^{\frac{1}{5}}\right)^{\frac{5}{2}}}$$ 4. **Multiply exponents inside numerator and denominator:** $$12^{\frac{1}{5} \times \frac{5}{2}} = 12^{\frac{1}{2}}$$ $$27^{\frac{1}{5} \times \frac{5}{2}} = 27^{\frac{1}{2}}$$ 5. **Rewrite the expression:** $$\frac{12^{\frac{1}{2}}}{27^{\frac{1}{2}}} = \frac{\sqrt{12}}{\sqrt{27}}$$ 6. **Simplify the square roots:** $$\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$$ $$\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}$$ 7. **Divide the simplified roots:** $$\frac{2\sqrt{3}}{3\sqrt{3}} = \frac{2}{3}$$ **Final answer:** $$\frac{2}{3}$$