e competencia 1 1. \( (2 x)(7 x+4)= \) Actividad de desarrollo 1. Realiza las siguientes multiplicaciones algebraicas y, en caso de ser necesario, reduce términos. g. \( \left(5 x^{3} y^{3}\right)\left(x^{2}-3 x y+y\right)= \) a. \( \left(6 x^{5} y^{6} z^{\prime}\right)\left(3 x^{2} y^{\prime} z^{\prime}\right)=1 \) h. \( \left(\frac{2 x^{4} y^{2}}{3}\right)\left(\frac{3 x^{2}}{2}+x^{2}-\frac{x y}{3}-\frac{2 x}{5}+2 x+5 y+12\right)= \) c. \( \left(\frac{2 x^{3} y^{1 / 2}}{3}\right)\left(\frac{3 x^{2} z^{4}}{5}\right)= \) i. \( (x-3)\left(x^{2}+3 x+9\right)= \)

Simplify the expression by following steps: - step0: Simplify the product: \(\left(x-3\right)\left(x^{2}+3x+9\right)\) - step1: Simplify the product: \(x^{3}-27\) Expand the expression \( (2 x)(7 x+4) \) Simplify the expression by following steps: - step0: Multiply the terms: \(\left(2x\right)\left(7x+4\right)\) - step1: Multiply the terms: \(2x\left(7x+4\right)\) - step2: Apply the distributive property: \(2x\times 7x+2x\times 4\) - step3: Multiply the terms: \(14x^{2}+8x\) Expand the expression \( (6 x^{5} y^{6} z')(3 x^{2} y' z') \) Simplify the expression by following steps: - step0: Simplify: \(\left(6x^{5}y^{6}\times \left(z\right)^{\prime}\right)\left(3x^{2}\times \left(y\right)^{\prime}\times \left(z\right)^{\prime}\right)\) - step1: Find the derivative: \(\left(6x^{5}y^{6}\times 1\right)\left(3x^{2}\times \left(y\right)^{\prime}\times \left(z\right)^{\prime}\right)\) - step2: Calculate: \(6x^{5}y^{6}\left(3x^{2}\times \left(y\right)^{\prime}\times \left(z\right)^{\prime}\right)\) - step3: Find the derivative: \(6x^{5}y^{6}\left(3x^{2}\times 1\times \left(z\right)^{\prime}\right)\) - step4: Find the derivative: \(6x^{5}y^{6}\left(3x^{2}\times 1\times 1\right)\) - step5: Multiply the terms: \(6x^{5}y^{6}\times 3x^{2}\) - step6: Multiply the numbers: \(18x^{5}y^{6}x^{2}\) - step7: Multiply the terms: \(18x^{7}y^{6}\) Expand the expression \( (5 x^{3} y^{3})(x^{2}-3 x y+y) \) Simplify the expression by following steps: - step0: Calculate: \(\left(5x^{3}y^{3}\right)\left(x^{2}-3xy+y\right)\) - step1: Calculate: \(5x^{3}y^{3}\left(x^{2}-3xy+y\right)\) - step2: Rewrite the expression: \(5\left(x^{2}-3xy+y\right)x^{3}y^{3}\) - step3: Rearrange the terms: \(\left(x^{2}-3xy+y\right)\times 5x^{3}y^{3}\) - step4: Apply the distributive property: \(x^{2}\times 5x^{3}y^{3}-3xy\times 5x^{3}y^{3}+y\times 5x^{3}y^{3}\) - step5: Multiply the terms: \(5x^{5}y^{3}-15x^{4}y^{4}+5y^{4}x^{3}\) Expand the expression \( (\frac{2 x^{3} y^{1 / 2}}{3})(\frac{3 x^{2} z^{4}}{5}) \) Simplify the expression by following steps: - step0: Simplify: \(\left(\frac{2x^{3}y^{\frac{1}{2}}}{3}\right)\left(\frac{3x^{2}z^{4}}{5}\right)\) - step1: Remove the parentheses: \(\frac{2x^{3}y^{\frac{1}{2}}}{3}\left(\frac{3x^{2}z^{4}}{5}\right)\) - step2: Remove the parentheses: \(\frac{2x^{3}y^{\frac{1}{2}}}{3}\times \frac{3x^{2}z^{4}}{5}\) - step3: Reduce the fraction: \(2x^{3}y^{\frac{1}{2}}\times \frac{x^{2}z^{4}}{5}\) - step4: Multiply the terms: \(\frac{2x^{3}y^{\frac{1}{2}}x^{2}z^{4}}{5}\) - step5: Multiply the terms: \(\frac{2x^{5}y^{\frac{1}{2}}z^{4}}{5}\) - step6: Simplify: \(\frac{2x^{5}\sqrt{y}\times z^{4}}{5}\) Expand the expression \( (\frac{2 x^{4} y^{2}}{3})(\frac{3 x^{2}}{2}+x^{2}-\frac{x y}{3}-\frac{2 x}{5}+2 x+5 y+12) \) Simplify the expression by following steps: - step0: Multiply the terms: \(\left(\frac{2x^{4}y^{2}}{3}\right)\left(\frac{3x^{2}}{2}+x^{2}-\frac{xy}{3}-\frac{2x}{5}+2x+5y+12\right)\) - step1: Remove the parentheses: \(\frac{2x^{4}y^{2}}{3}\left(\frac{3x^{2}}{2}+x^{2}-\frac{xy}{3}-\frac{2x}{5}+2x+5y+12\right)\) - step2: Calculate: \(\frac{2x^{4}y^{2}}{3}\times \frac{75x^{2}-10xy+48x+150y+360}{30}\) - step3: Reduce the fraction: \(\frac{x^{4}y^{2}}{3}\times \frac{75x^{2}-10xy+48x+150y+360}{15}\) - step4: Multiply the terms: \(\frac{x^{4}y^{2}\left(75x^{2}-10xy+48x+150y+360\right)}{3\times 15}\) - step5: Multiply the terms: \(\frac{x^{4}y^{2}\left(75x^{2}-10xy+48x+150y+360\right)}{45}\) - step6: Multiply the terms: \(\frac{75x^{6}y^{2}-10x^{5}y^{3}+48x^{5}y^{2}+150x^{4}y^{3}+360x^{4}y^{2}}{45}\) Aquí están las soluciones a las multiplicaciones algebraicas solicitadas: 1. **Multiplicación de \( (2 x)(7 x+4) \)**: \[ (2 x)(7 x + 4) = 14x^{2} + 8x \] 2. **Multiplicación de \( (5 x^{3} y^{3})(x^{2}-3 x y+y) \)**: \[ (5 x^{3} y^{3})(x^{2} - 3 x y + y) = 5x^{5}y^{3} - 15x^{4}y^{4} + 5y^{4}x^{3} \] 3. **Multiplicación de \( (6 x^{5} y^{6} z')(3 x^{2} y' z') \)**: \[ (6 x^{5} y^{6} z')(3 x^{2} y' z') = 18x^{7}y^{6} \] 4. **Multiplicación de \( \left(\frac{2 x^{4} y^{2}}{3}\right)\left(\frac{3 x^{2}}{2}+x^{2}-\frac{x y}{3}-\frac{2 x}{5}+2 x+5 y+12\right) \)**: \[ \left(\frac{2 x^{4} y^{2}}{3}\right)\left(\frac{3 x^{2}}{2}+x^{2}-\frac{x y}{3}-\frac{2 x}{5}+2 x+5 y+12\right) = \frac{75x^{6}y^{2}-10x^{5}y^{3}+48x^{5}y^{2}+150x^{4}y^{3}+360x^{4}y^{2}}{45} \] 5. **Multiplicación de \( \left(\frac{2 x^{3} y^{1 / 2}}{3}\right)\left(\frac{3 x^{2} z^{4}}{5}\right) \)**: \[ \left(\frac{2 x^{3} y^{1 / 2}}{3}\right)\left(\frac{3 x^{2} z^{4}}{5}\right) = \frac{2x^{5}\sqrt{y}\times z^{4}}{5} \] 6. **Multiplicación de \( (x-3)(x^{2}+3 x+9) \)**: \[ (x-3)(x^{2}+3 x+9) = x^{3} - 27 \] Si necesitas más ayuda o aclaraciones sobre algún paso, no dudes en preguntar.