1. The problem involves simplifying expressions with rational exponents and radicals. 2. Recall the key rule: $$a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$$ where $a$ is the base, $m$ is the numerator exponent, and $n$ is the denominator root. 3. To simplify, convert radicals to rational exponents or vice versa, then apply exponent rules such as multiplication, division, and power of a power. 4. For example, simplify $\sqrt[3]{x^6}$: $$\sqrt[3]{x^6} = x^{\frac{6}{3}} = x^2$$ 5. Another example, simplify $\left( x^{\frac{1}{2}} \right)^4$: $$\left( x^{\frac{1}{2}} \right)^4 = x^{\frac{1}{2} \times 4} = x^2$$ 6. When multiplying expressions with the same base, add exponents: $$x^a \times x^b = x^{a+b}$$ 7. When dividing, subtract exponents: $$\frac{x^a}{x^b} = x^{a-b}$$ 8. Simplify each puzzle piece expression using these rules, convert all answers to simplest radical form if needed. 9. Arrange puzzle pieces so that edges with matching simplified expressions align. 10. This process helps reinforce understanding of rational exponents and radicals through practice and pattern recognition.