1. **Problem:** Simplify the expression $$\frac{(2 x^3 y^{-2})^3 \times (4 x^{\frac{1}{2}} y^5)}{8 x^2 y^3}$$ 2. **Formula and rules:** Use the power of a product rule: $$(ab)^n = a^n b^n$$ Use the power of a power rule: $$\left(x^a\right)^b = x^{ab}$$ Use multiplication of powers with the same base: $$x^a \times x^b = x^{a+b}$$ Use division of powers with the same base: $$\frac{x^a}{x^b} = x^{a-b}$$ 3. **Step-by-step simplification:** - Expand powers: $$(2 x^3 y^{-2})^3 = 2^3 x^{3 \times 3} y^{-2 \times 3} = 8 x^9 y^{-6}$$ - Multiply by $4 x^{\frac{1}{2}} y^5$: $$8 x^9 y^{-6} \times 4 x^{\frac{1}{2}} y^5 = 32 x^{9 + \frac{1}{2}} y^{-6 + 5} = 32 x^{\frac{19}{2}} y^{-1}$$ - Divide by denominator: $$\frac{32 x^{\frac{19}{2}} y^{-1}}{8 x^2 y^3} = 4 x^{\frac{19}{2} - 2} y^{-1 - 3} = 4 x^{\frac{15}{2}} y^{-4}$$ 4. **Final answer:** $$4 x^{\frac{15}{2}} y^{-4}$$ --- Since the user asked to simplify all questions but per instructions only the first problem is solved completely. "slug": "exponent simplification", "subject": "algebra", "desmos": {"latex": "y=4 x^{15/2} y^{-4}", "features": {"intercepts": true, "extrema": true}}, "q_count": 6