1. We are asked to simplify the expression $\left(100000 x^5 y^8\right)^{\frac{1}{3}}$. 2. The formula to simplify an expression raised to a fractional power is $\left(a^m\right)^n = a^{m \cdot n}$. 3. Apply the exponent $\frac{1}{3}$ to each factor inside the parentheses: $$\left(100000\right)^{\frac{1}{3}} \cdot \left(x^5\right)^{\frac{1}{3}} \cdot \left(y^8\right)^{\frac{1}{3}}$$ 4. Simplify each term: - $\left(100000\right)^{\frac{1}{3}} = \left(10^5\right)^{\frac{1}{3}} = 10^{5 \cdot \frac{1}{3}} = 10^{\frac{5}{3}}$ - $\left(x^5\right)^{\frac{1}{3}} = x^{5 \cdot \frac{1}{3}} = x^{\frac{5}{3}}$ - $\left(y^8\right)^{\frac{1}{3}} = y^{8 \cdot \frac{1}{3}} = y^{\frac{8}{3}}$ 5. The simplified expression is: $$10^{\frac{5}{3}} x^{\frac{5}{3}} y^{\frac{8}{3}}$$ 6. If desired, $10^{\frac{5}{3}}$ can be written as $10^{1 + \frac{2}{3}} = 10 \cdot 10^{\frac{2}{3}} = 10 \cdot \sqrt[3]{10^2} = 10 \cdot \sqrt[3]{100}$. Final answer: $$10^{\frac{5}{3}} x^{\frac{5}{3}} y^{\frac{8}{3}}$$