1. **Simplify each expression and leave in fraction and exponent form with positive exponents.** --- a) Simplify $$\frac{((-2x^{-3}y)^{-2}(-12x^{-4}y^{-2}))}{6xy^{-3}}$$ Step 1: Apply the negative exponent rule $$a^{-n} = \frac{1}{a^n}$$ to $$(-2x^{-3}y)^{-2}$$: $$(-2x^{-3}y)^{-2} = \frac{1}{(-2x^{-3}y)^2} = \frac{1}{(-2)^2 (x^{-3})^2 y^2} = \frac{1}{4 x^{-6} y^2}$$ Step 2: Rewrite $$x^{-6}$$ as $$\frac{1}{x^6}$$: $$\frac{1}{4 x^{-6} y^2} = \frac{1}{4 \cdot \frac{1}{x^6} \cdot y^2} = \frac{1}{4} \cdot x^6 \cdot \frac{1}{y^2} = \frac{x^6}{4 y^2}$$ Step 3: Multiply by $$(-12 x^{-4} y^{-2})$$: $$\frac{x^6}{4 y^2} \times (-12 x^{-4} y^{-2}) = \frac{x^6}{4 y^2} \times (-12) \times x^{-4} \times y^{-2}$$ Step 4: Combine like bases by adding exponents: $$x^{6 + (-4)} = x^2$$ $$y^{(-2) + (-2)} = y^{-4}$$ Step 5: Multiply constants: $$\frac{1}{4} \times (-12) = -3$$ So the numerator becomes: $$-3 x^2 y^{-4}$$ Step 6: Divide by denominator $$6 x y^{-3}$$: $$\frac{-3 x^2 y^{-4}}{6 x y^{-3}}$$ Step 7: Simplify constants: $$\frac{-3}{6} = -\frac{1}{2}$$ Step 8: Simplify $$x$$ terms: $$\frac{x^2}{x} = x^{2-1} = x^1 = x$$ Step 9: Simplify $$y$$ terms: $$\frac{y^{-4}}{y^{-3}} = y^{-4 - (-3)} = y^{-1} = \frac{1}{y}$$ Step 10: Final expression: $$-\frac{1}{2} x \cdot \frac{1}{y} = -\frac{x}{2 y}$$ --- b) Simplify $$((a^{-4} b^{-8})^{\frac{1}{2}} (a^{6} b^{9})^{\frac{1}{3}})$$ Step 1: Apply power to power rule $$ (x^m)^n = x^{m n} $$: $$ (a^{-4} b^{-8})^{\frac{1}{2}} = a^{-4 \times \frac{1}{2}} b^{-8 \times \frac{1}{2}} = a^{-2} b^{-4} $$ $$ (a^{6} b^{9})^{\frac{1}{3}} = a^{6 \times \frac{1}{3}} b^{9 \times \frac{1}{3}} = a^{2} b^{3} $$ Step 2: Multiply the two expressions: $$ a^{-2} b^{-4} \times a^{2} b^{3} = a^{-2 + 2} b^{-4 + 3} = a^{0} b^{-1} = b^{-1} = \frac{1}{b} $$ --- c) Simplify $$ (27 x^{9})^{-\frac{1}{3}} $$ Step 1: Apply power to power rule: $$ 27^{-\frac{1}{3}} x^{9 \times (-\frac{1}{3})} = 27^{-\frac{1}{3}} x^{-3} $$ Step 2: Simplify $$27^{-\frac{1}{3}}$$: Since $$27 = 3^3$$, then $$27^{-\frac{1}{3}} = (3^3)^{-\frac{1}{3}} = 3^{3 \times (-\frac{1}{3})} = 3^{-1} = \frac{1}{3}$$ Step 3: Final expression: $$ \frac{1}{3} x^{-3} = \frac{1}{3 x^{3}} $$ --- d) Simplify $$ \frac{(-4 s^{-2} t^{-3})^{2}}{- s^{2} t^{-1}} $$ Step 1: Square the numerator: $$ (-4)^{2} (s^{-2})^{2} (t^{-3})^{2} = 16 s^{-4} t^{-6} $$ Step 2: Write the expression: $$ \frac{16 s^{-4} t^{-6}}{- s^{2} t^{-1}} $$ Step 3: Simplify constants: $$ \frac{16}{-1} = -16 $$ Step 4: Simplify $$s$$ terms: $$ s^{-4 - 2} = s^{-6} = \frac{1}{s^{6}} $$ Step 5: Simplify $$t$$ terms: $$ t^{-6 - (-1)} = t^{-6 + 1} = t^{-5} = \frac{1}{t^{5}} $$ Step 6: Final expression: $$ -16 \cdot \frac{1}{s^{6}} \cdot \frac{1}{t^{5}} = -\frac{16}{s^{6} t^{5}} $$ --- e) Simplify $$ \sqrt{6 \sqrt{6}} $$ Step 1: Rewrite inner root: $$ \sqrt{6} = 6^{\frac{1}{2}} $$ Step 2: Expression becomes: $$ \sqrt{6 \times 6^{\frac{1}{2}}} = \sqrt{6^{1 + \frac{1}{2}}} = \sqrt{6^{\frac{3}{2}}} $$ Step 3: Simplify outer root: $$ (6^{\frac{3}{2}})^{\frac{1}{2}} = 6^{\frac{3}{2} \times \frac{1}{2}} = 6^{\frac{3}{4}} $$ --- f) Simplify $$ \sqrt[3]{5 \sqrt{5 (5^{3})}} $$ Step 1: Simplify inside the square root: $$ 5^{3} = 125 $$ So inside the square root: $$ 5 \times 125 = 5^{1} \times 5^{3} = 5^{4} $$ Step 2: Square root of $$5^{4}$$: $$ \sqrt{5^{4}} = 5^{\frac{4}{2}} = 5^{2} = 25 $$ Step 3: Expression becomes: $$ \sqrt[3]{5 \times 25} = \sqrt[3]{125} $$ Step 4: Simplify cube root: $$ \sqrt[3]{125} = 5 $$ --- **Final answers:** a) $$-\frac{x}{2 y}$$ b) $$\frac{1}{b}$$ c) $$\frac{1}{3 x^{3}}$$ d) $$-\frac{16}{s^{6} t^{5}}$$ e) $$6^{\frac{3}{4}}$$ f) $$5$$