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. $$ (x-3)\left(x^{2}+3 x+9\right)= $$

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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.
Aquí están las soluciones a las multiplicaciones algebraicas: 1. $$ (2x)(7x + 4) = 14x^{2} + 8x $$ 2. $$ (5x^{3}y^{3})(x^{2} - 3xy + y) = 5x^{5}y^{3} - 15x^{4}y^{4} + 5y^{4}x^{3} $$ 3. $$ (6x^{5}y^{6}z')(3x^{2}y'z') = 18x^{7}y^{6} $$ 4. $$ \left(\frac{2x^{4}y^{2}}{3}\right)\left(\frac{3x^{2}}{2} + x^{2} - \frac{xy}{3} - \frac{2x}{5} + 2x + 5y + 12\right) = \frac{75x^{6}y^{2} - 10x^{5}y^{3} + 48x^{5}y^{2} + 150x^{4}y^{3} + 360x^{4}y^{2}}{45} $$ 5. $$ \left(\frac{2x^{3}y^{1/2}}{3}\right)\left(\frac{3x^{2}z^{4}}{5}\right) = \frac{2x^{5}\sqrt{y} \times z^{4}}{5} $$ 6. $$ (x - 3)(x^{2} + 3x + 9) = x^{3} - 27 $$ Si tienes más preguntas o necesitas aclaraciones, no dudes en preguntar.

e competencia 1

Actividad de desarrollo

Realiza las siguientes multiplicaciones algebraicas y, en caso de ser necesario, reduce términos.
g.
a.
h.
c.
i.

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e competencia 1 Actividad de desarrollo Realiza las siguientes multiplicaciones algebraicas y, en caso de ser necesario, reduce términos. g. a. h. c. i.