(a) EJtRCTACIÓN. Calcular las siguientes potencias. Expre sar las respirestas conn exponentes positivos. 1. \( (9 x)^{-4} \) 4. \( \left(-1,1 x^{2} y^{4}\right)^{2} \) 2. \( \left(-4 a^{3} b^{3} c^{2}\right)^{3} \) 3. \( \left(0,5, y^{5}\right)^{3} \) 5. \( (2 \pi r)^{3} \) 7. \( \left(\frac{7 a b^{3}}{3 c^{2}}\right)^{-2} \) 6. \( \left(-3 a^{5} b^{3} c^{6} d^{4}\right)^{0} \) 8. \( \left(\frac{3}{4} m^{2} n^{-3}\right)^{2} \) \( 9 .\left(\frac{4}{3} \pi r^{3}\right)^{-3} \) (2) RAZONAMIENTO. Marcar con \( V \), si la afirmación es verdadera o wi es falsa. Juistificar la respuesta. 10. \( -(x y)^{-2}=\frac{x^{2}}{y^{2}} \) 11. \( -(x \sqrt{2})^{0}=-\sqrt{2} \) 12. \( -\left(w_{y^{2}}^{2}\right)^{3}=\left(w y^{3}\right)^{2} \) 13. \( -8 a^{3} b^{3}=(2 a b)^{3} \) 14. \( -\left(-\frac{7}{5} x^{-2} y^{3}\right)^{2}=-\frac{49 y^{3}}{25 x^{4}} \)

Simplify the expression by following steps: - step0: Solution: \(\left(-3a^{5}b^{3}c^{6}d^{4}\right)^{0}\) - step1: Simplify: \(1\) Calculate or simplify the expression \( (2 * \pi * r)^3 \). Simplify the expression by following steps: - step0: Solution: \(\left(2\pi r\right)^{3}\) - step1: Use the properties of exponents: \(\left(2\pi \right)^{3}r^{3}\) - step2: Evaluate the power: \(8\pi ^{3}r^{3}\) Calculate or simplify the expression \( (4/3 * \pi * r^3)^-3 \). Simplify the expression by following steps: - step0: Solution: \(\left(\frac{4}{3}\pi r^{3}\right)-3\) - step1: Multiply: \(\frac{4\pi }{3}r^{3}-3\) Calculate or simplify the expression \( (-4 * a^3 * b^3 * c^2)^3 \). Simplify the expression by following steps: - step0: Solution: \(\left(-4a^{3}b^{3}c^{2}\right)^{3}\) - step1: Determine the sign: \(-\left(4a^{3}b^{3}c^{2}\right)^{3}\) - step2: Use the properties of exponents: \(-4^{3}\left(a^{3}\right)^{3}\left(b^{3}\right)^{3}\left(c^{2}\right)^{3}\) - step3: Evaluate the power: \(-64a^{9}b^{9}c^{6}\) Calculate or simplify the expression \( (9 * x)^-4 \). Simplify the expression by following steps: - step0: Solution: \(9x-4\) Calculate or simplify the expression \( (7 * a * b^3)/(3 * c^2) \). Simplify the expression by following steps: - step0: Solution: \(\frac{7ab^{3}}{3c^{2}}\) Calculate or simplify the expression \( (0.5 * y^5)^3 \). Simplify the expression by following steps: - step0: Solution: \(\left(0.5y^{5}\right)^{3}\) - step1: Convert the expressions: \(\left(\frac{1}{2}y^{5}\right)^{3}\) - step2: Use the properties of exponents: \(\left(\frac{1}{2}\right)^{3}\left(y^{5}\right)^{3}\) - step3: Evaluate the power: \(\frac{1}{8}y^{15}\) Calculate or simplify the expression \( (3/4 * m^2 * n^-3)^2 \). Simplify the expression by following steps: - step0: Solution: \(\left(\frac{3}{4}m^{2}n-3\right)^{2}\) - step1: Calculate: \(\frac{9}{16}m^{4}n^{2}-\frac{9}{2}m^{2}n+9\) Calculate or simplify the expression \( -(x * y)^-2 \). Simplify the expression by following steps: - step0: Solution: \(-xy-2\) Calculate or simplify the expression \( -(x * \sqrt(2))^0 \). Simplify the expression by following steps: - step0: Solution: \(-\left(x\sqrt{2}\right)^{0}\) - step1: Calculate: \(-\left(\sqrt{2}\times x\right)^{0}\) - step2: Simplify: \(-1\) Calculate or simplify the expression \( -((w * y^2)^2)^3 \). Simplify the expression by following steps: - step0: Solution: \(-\left(\left(wy^{2}\right)^{2}\right)^{3}\) - step1: Multiply the exponents: \(-\left(wy^{2}\right)^{2\times 3}\) - step2: Multiply the numbers: \(-\left(wy^{2}\right)^{6}\) - step3: Evaluate the power: \(-w^{6}y^{12}\) Calculate or simplify the expression \( -8 * a^3 * b^3 \). Simplify the expression by following steps: - step0: Solution: \(-8a^{3}b^{3}\) Calculate or simplify the expression \( -((-7/5) * x^-2 * y^3)^2 \). Simplify the expression by following steps: - step0: Solution: \(-\left(\left(\frac{-7}{5}\right)x-2y^{3}\right)^{2}\) - step1: Rewrite the fraction: \(-\left(\left(-\frac{7}{5}\right)x-2y^{3}\right)^{2}\) - step2: Remove the parentheses: \(-\left(-\frac{7}{5}x-2y^{3}\right)^{2}\) - step3: Evaluate the power: \(-\left(\frac{7}{5}x+2y^{3}\right)^{2}\) - step4: Calculate: \(-\left(\frac{49}{25}x^{2}+\frac{28}{5}xy^{3}+4y^{6}\right)\) - step5: Calculate: \(-\frac{49}{25}x^{2}-\frac{28}{5}xy^{3}-4y^{6}\) Aquí están las soluciones a las potencias y razonamientos solicitados: ### (a) Cálculo de potencias 1. \( (9 x)^{-4} = \frac{1}{(9 x)^{4}} = \frac{1}{9^{4} x^{4}} = \frac{1}{6561 x^{4}} \) 2. \( \left(-4 a^{3} b^{3} c^{2}\right)^{3} = -64 a^{9} b^{9} c^{6} \) 3. \( \left(0.5 y^{5}\right)^{3} = \frac{1}{8} y^{15} \) 4. \( (2 \pi r)^{3} = 8 \pi^{3} r^{3} \) 5. \( (2 \pi r)^{3} = 8 \pi^{3} r^{3} \) 6. \( \left(-3 a^{5} b^{3} c^{6} d^{4}\right)^{0} = 1 \) (cualquier número elevado a la potencia 0 es 1) 7. \( \left(\frac{7 a b^{3}}{3 c^{2}}\right)^{-2} = \frac{9 c^{4}}{49 a^{2} b^{6}} \) 8. \( \left(\frac{3}{4} m^{2} n^{-3}\right)^{2} = \frac{9}{16} m^{4} n^{2} \) 9. \( \left(\frac{4}{3} \pi r^{3}\right)^{-3} = \frac{27}{64 \pi^{3} r^{9}} \) ### (2) Razonamiento 10. \( -(x y)^{-2} = -\frac{1}{(xy)^{2}} = -\frac{x^{2}}{y^{2}} \) → **Falso** (la expresión correcta es \(-\frac{1}{x^{2}y^{2}}\)) 11. \( -(x \sqrt{2})^{0} = -1 \) → **Verdadero** (cualquier número elevado a 0 es 1) 12. \( -\left(w y^{2}\right)^{6} = -w^{6} y^{12} \) → **Verdadero** 13. \( -8 a^{3} b^{3} = -(2 a b)^{3} \) → **Falso** (la expresión correcta es \(-8 a^{3} b^{3}\)) 14. \( -\left(-\frac{7}{5} x^{-2} y^{3}\right)^{2} = -\frac{49 y^{6}}{25 x^{4}} \) → **Verdadero** Si necesitas más aclaraciones o cálculos adicionales, no dudes en preguntar.