1. Problem a: Simplify $a^{3/2} \cdot a^{4/3}$. Use the exponent multiplication rule $a^m \cdot a^n = a^{m+n}$: $$ a^{3/2} \cdot a^{4/3} = a^{3/2 + 4/3}. $$ Find a common denominator for the exponents 2 and 3 which is 6: $$ \frac{3}{2} = \frac{9}{6}, \quad \frac{4}{3} = \frac{8}{6}. $$ Sum: $$ \frac{9}{6} + \frac{8}{6} = \frac{17}{6}. $$ Answer: $$ a^{17/6}. $$ 2. Problem b: Simplify $(-3x^{2/3})^3$. Apply the power of a product rule: $$ (-3)^3 \cdot \left(x^{2/3}\right)^3 = -27 \cdot x^{2/3 \times 3} = -27x^2. $$ 3. Problem c: Simplify $(x^{2/3} \cdot x^{4/5})^2$. Inside the parentheses, add exponents: $$ x^{2/3 + 4/5}. $$ Common denominator 15: $$ \frac{2}{3} = \frac{10}{15}, \quad \frac{4}{5} = \frac{12}{15}. $$ Sum: $$ \frac{10}{15} + \frac{12}{15} = \frac{22}{15}. $$ Raise to power 2: $$ \left(x^{22/15}\right)^2 = x^{44/15}. $$ 4. Problem d: Simplify $(x^{2/5} \cdot x^{1/3}) \div x^{3/5}$. Multiply exponents inside numerator: $$ x^{2/5 + 1/3}. $$ Common denominator 15: $$ \frac{2}{5} = \frac{6}{15}, \quad \frac{1}{3} = \frac{5}{15}. $$ Sum: $$ \frac{6}{15} + \frac{5}{15} = \frac{11}{15}. $$ Divide by $x^{3/5} = x^{9/15}$: $$ x^{11/15 - 9/15} = x^{2/15}. $$ 5. Problem e: Simplify $\sqrt[3]{-8x^6}$. Rewrite as cube roots of factors: $$ \sqrt[3]{-8} \cdot \sqrt[3]{x^6} = -2 \cdot x^{6/3} = -2x^2. $$ 6. Problem f: Simplify $\sqrt[4]{1296 a^8}$. Express as fourth roots of each factor: $$ \sqrt[4]{1296} \cdot \sqrt[4]{a^8}. $$ Since $1296 = 6^4$, $$ \sqrt[4]{1296} = 6. $$ $$ \sqrt[4]{a^8} = a^{8/4} = a^2. $$ Answer: $$ 6a^2. $$ 7. Problem g: Simplify $\sqrt{2} \cdot \sqrt{8}$. Multiply radicands: $$ \sqrt{2 \times 8} = \sqrt{16} = 4. $$ 8. Problem h: Simplify $\frac{\sqrt{2} \cdot \sqrt{8} \cdot \sqrt{4}}{\sqrt[3]{32} \cdot \sqrt[3]{2}}$. Numerator: $$ \sqrt{2 \times 8 \times 4} = \sqrt{64} = 8. $$ Denominator: $$ \sqrt[3]{32} \cdot \sqrt[3]{2} = \sqrt[3]{32 \times 2} = \sqrt[3]{64} = 4. $$ Result: $$ \frac{8}{4} = 2. $$ 9. Problem i: Simplify $\sqrt[3]{x^5}$. Rewrite exponent: $$ x^{5/3}. $$ 10. Problem j: Simplify $\sqrt[3]{-\frac{1}{125}}$. Cube root applies to numerator and denominator: $$ \frac{\sqrt[3]{-1}}{\sqrt[3]{125}} = \frac{-1}{5} = -\frac{1}{5}. $$ Final answers summary: a) $a^{17/6}$ b) $-27x^2$ c) $x^{44/15}$ d) $x^{2/15}$ e) $-2x^2$ f) $6a^2$ g) $4$ h) $2$ i) $x^{5/3}$ j) $-\frac{1}{5}$