1. **Problem Statement:** Translate the given expressions between exponential and radical forms. 2. **Key Formula:** The relationship between radicals and rational exponents is: $$\sqrt[n]{a} = a^{\frac{1}{n}}$$ and conversely, $$a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m$$ 3. **Part A: Convert to Radical Form** - Expression 1: $5^{\frac{1}{6}}$ Using the formula, this is the sixth root of 5: $$5^{\frac{1}{6}} = \sqrt[6]{5}$$ - Expression 2: $-12(xyz^9)^{\frac{1}{8}}$ Rewrite the exponent as an eighth root: $$-12 \times \sqrt[8]{xyz^9}$$ Since $z^9$ is inside the root, it remains as is. 4. **Part B: Convert to Exponential Form** - Expression 3: $\sqrt[3]{2}$ This is the cube root of 2, which equals: $$2^{\frac{1}{3}}$$ - Expression 4: $\sqrt[4]{\frac{5}{y^3}}$ Rewrite the fourth root as an exponent: $$\left(\frac{5}{y^3}\right)^{\frac{1}{4}} = \frac{5^{\frac{1}{4}}}{y^{\frac{3}{4}}}$$ 5. **Summary:** - $5^{\frac{1}{6}} = \sqrt[6]{5}$ - $-12(xyz^9)^{\frac{1}{8}} = -12 \sqrt[8]{xyz^9}$ - $\sqrt[3]{2} = 2^{\frac{1}{3}}$ - $\sqrt[4]{\frac{5}{y^3}} = \frac{5^{\frac{1}{4}}}{y^{\frac{3}{4}}}$ This completes the conversions between radical and exponential forms.