Ejercitación (2. Determina el valor numérico de las siguientes ex- rresiones algebraicas, sabiendo que $$ x=-2, y=3 $$ d. $$ \frac{2}{3} x^{3} y^{2} z-5 x^{2} y^{3} z^{2}+10 $$ e. $$ \frac{3}{4} x y^{2} z^{3}-x^{2} y^{3} z^{2}+x^{3} y^{2} z^{3}-\frac{1}{2} $$

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Evaluate the expression by following steps:
- step0: Evaluate:
  $$\frac{2}{3}x^{3}y^{2}z-5x^{2}y^{3}z^{2}+10$$
- step1: Substitute:
  $$\frac{2}{3}\left(-2\right)^{3}\times 3^{2}\times 1-5\left(-2\right)^{2}\times 3^{3}\times 1^{2}+10$$
- step2: Evaluate the power:
  $$\frac{2}{3}\left(-2\right)^{3}\times 3^{2}\times 1-5\left(-2\right)^{2}\times 3^{3}\times 1+10$$
- step3: Multiply the terms:
  $$-48-5\left(-2\right)^{2}\times 3^{3}\times 1+10$$
- step4: Multiply the terms:
  $$-48-540+10$$
- step5: Calculate:
  $$-578$$
Substitute $$ x=-2,y=3,z=1 $$ into the expression $$ \frac{3}{4} x y^{2} z^{3}-x^{2} y^{3} z^{2}+x^{3} y^{2} z^{3}-\frac{1}{2} $$.
Evaluate the expression by following steps:
- step0: Evaluate:
  $$\frac{3}{4}xy^{2}z^{3}-x^{2}y^{3}z^{2}+x^{3}y^{2}z^{3}-\frac{1}{2}$$
- step1: Substitute:
  $$\frac{3}{4}\left(-2\right)\times 3^{2}\times 1^{3}-\left(-2\right)^{2}\times 3^{3}\times 1^{2}+\left(-2\right)^{3}\times 3^{2}\times 1^{3}-\frac{1}{2}$$
- step2: Evaluate the power:
  $$\frac{3}{4}\left(-2\right)\times 3^{2}\times 1-\left(-2\right)^{2}\times 3^{3}\times 1^{2}+\left(-2\right)^{3}\times 3^{2}\times 1^{3}-\frac{1}{2}$$
- step3: Evaluate the power:
  $$\frac{3}{4}\left(-2\right)\times 3^{2}\times 1-\left(-2\right)^{2}\times 3^{3}\times 1+\left(-2\right)^{3}\times 3^{2}\times 1^{3}-\frac{1}{2}$$
- step4: Evaluate the power:
  $$\frac{3}{4}\left(-2\right)\times 3^{2}\times 1-\left(-2\right)^{2}\times 3^{3}\times 1+\left(-2\right)^{3}\times 3^{2}\times 1-\frac{1}{2}$$
- step5: Multiply the terms:
  $$-\frac{3^{3}}{2}-\left(-2\right)^{2}\times 3^{3}\times 1+\left(-2\right)^{3}\times 3^{2}\times 1-\frac{1}{2}$$
- step6: Multiply the terms:
  $$-\frac{3^{3}}{2}-108+\left(-2\right)^{3}\times 3^{2}\times 1-\frac{1}{2}$$
- step7: Multiply the terms:
  $$-\frac{3^{3}}{2}-108-72-\frac{1}{2}$$
- step8: Evaluate the power:
  $$-\frac{27}{2}-108-72-\frac{1}{2}$$
- step9: Subtract the numbers:
  $$-\frac{27}{2}-180-\frac{1}{2}$$
- step10: Reduce fractions to a common denominator:
  $$-\frac{27}{2}-\frac{180\times 2}{2}-\frac{1}{2}$$
- step11: Transform the expression:
  $$\frac{-27-180\times 2-1}{2}$$
- step12: Multiply the numbers:
  $$\frac{-27-360-1}{2}$$
- step13: Subtract the numbers:
  $$\frac{-388}{2}$$
- step14: Reduce the numbers:
  $$\frac{-194}{1}$$
- step15: Calculate:
  $$-194$$
Vamos a determinar el valor numérico de las expresiones algebraicas dadas, utilizando los valores $$ x = -2 $$, $$ y = 3 $$ y $$ z = 1 $$ (asumimos $$ z = 1 $$ ya que no se proporcionó un valor para $$ z $$).

### Parte d
La expresión es:
$$
\frac{2}{3} x^{3} y^{2} z - 5 x^{2} y^{3} z^{2} + 10
$$

Sustituyendo los valores:
$$
\frac{2}{3} (-2)^{3} (3)^{2} (1) - 5 (-2)^{2} (3)^{3} (1)^{2} + 10
$$

Calculamos cada término:
1. $$ (-2)^{3} = -8 $$
2. $$ (3)^{2} = 9 $$
3. $$ (-2)^{2} = 4 $$
4. $$ (3)^{3} = 27 $$

Sustituyendo estos valores:
$$
\frac{2}{3} (-8) (9) (1) - 5 (4) (27) (1) + 10
$$
$$
= \frac{2}{3} (-72) - 5 (108) + 10
$$
$$
= -48 - 540 + 10
$$
$$
= -578
$$

### Parte e
La expresión es:
$$
\frac{3}{4} x y^{2} z^{3} - x^{2} y^{3} z^{2} + x^{3} y^{2} z^{3} - \frac{1}{2}
$$

Sustituyendo los valores:
$$
\frac{3}{4} (-2) (3)^{2} (1)^{3} - (-2)^{2} (3)^{3} (1)^{2} + (-2)^{3} (3)^{2} (1)^{3} - \frac{1}{2}
$$

Calculamos cada término:
1. $$ (3)^{2} = 9 $$
2. $$ (-2)^{2} = 4 $$
3. $$ (3)^{3} = 27 $$
4. $$ (-2)^{3} = -8 $$

Sustituyendo estos valores:
$$
\frac{3}{4} (-2) (9) (1) - (4) (27) (1) + (-8) (9) (1) - \frac{1}{2}
$$
$$
= \frac{3}{4} (-18) - 108 - 72 - \frac{1}{2}
$$
$$
= -13.5 - 108 - 72 - 0.5
$$
$$
= -194
$$

### Resultados Finales
- Para la parte d: $$ -578 $$
- Para la parte e: $$ -194 $$
Para $$ x = -2 $$ y $$ y = 3 $$: - **Parte d**: $$ -578 $$ - **Parte e**: $$ -194 $$

Ejercitación (2. Determina el valor numérico de las siguientes ex- rresiones algebraicas, sabiendo que \( x=-2, y=3 \) d. \( \frac{2}{3} x^{3} y^{2} z-5 x^{2} y^{3} z^{2}+10 \) e. \( \frac{3}{4} x y^{2} z^{3}-x^{2} y^{3} z^{2}+x^{3} y^{2} z^{3}-\frac{1}{2} \)

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