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Sagot :
Para resolver cada una de las expresiones dadas paso a paso, procederemos a simplificarlas ordenadamente.
Comencemos con los valores de las constantes importantes involucradas:
- \(\pi \approx 3.14159\)
- \(e \approx 2.7183\)
- \(\phi = \frac{1 + \sqrt{5}}{2} \approx 1.618\)
### 1) Simplificación de:
[tex]\[ \left[\sqrt[3]{24} \times\left\{-5^4 \div\left[2-\left(\frac{1}{3-5 \frac{7}{9}}\right)\right]\right\}-\left(3 \frac{5}{7}+5\right) \times\left(-9 \frac{5}{7} \div 2 \frac{3}{5}\right)\right] \][/tex]
#### Paso 1:
Calculamos \(\sqrt[3]{24}\):
[tex]\[ \sqrt[3]{24} \approx 2.884 \][/tex]
#### Paso 2:
Simplificamos el denominador complejo del primer término:
[tex]\[ 5 \frac{7}{9} = \frac{5 \cdot 7}{9} = \frac{35}{9} \][/tex]
[tex]\[ 3 - \frac{35}{9} = \frac{27}{9} - \frac{35}{9} = \frac{27 - 35}{9} = \frac{-8}{9} \][/tex]
[tex]\[ \frac{1}{\frac{-8}{9}} = -\frac{9}{8} \][/tex]
[tex]\[ 2 - \left(-\frac{9}{8}\right) = 2 + \frac{9}{8} = \frac{16}{8} + \frac{9}{8} = \frac{25}{8} \][/tex]
#### Paso 3:
Simplificamos el término en el numerador:
[tex]\[ -5^4 = -625 \][/tex]
[tex]\[ -625 \div \frac{25}{8} = -625 \times \frac{8}{25} = -25 \times 8 = -200 \][/tex]
#### Paso 4:
El primer término queda:
[tex]\[ \sqrt[3]{24} \times -200 \approx 2.884 \times -200 = -576.8 \][/tex]
#### Paso 5:
Simplificamos el segundo término:
[tex]\[ 3 \frac{5}{7} + 5 = \frac{3 \cdot 5}{7} + 5 = \frac{15}{7} + 5 = \frac{15 + 35}{7} = \frac{50}{7} \][/tex]
[tex]\[ -9 \frac{5}{7} = -\frac{9 \cdot 5}{7} = -\frac{45}{7} \][/tex]
[tex]\[ 2 \frac{3}{5} = \frac{2 \cdot 3}{5} = \frac{6}{5} \][/tex]
[tex]\[ -\frac{45}{7} \div \frac{6}{5} = -\frac{45}{7} \times \frac{5}{6} = -\frac{225}{42} = -\frac{75}{14} \][/tex]
[tex]\[ \text{Segundo término} = \frac{50}{7} \times -\frac{75}{14} = \frac{50 \cdot -75}{7 \cdot 14} = -\frac{3750}{98} \approx -38.27 \][/tex]
#### Resolución del primer punto:
[tex]\[ -576.8 - (-38.27) = -576.8 + 38.27 = -538.53 \][/tex]
### 2) Simplificación de:
[tex]\[ \left(\left\{\frac{e}{3}[2 \phi]-\left(-\frac{5}{9}\right)^3\right\}+\left[\pi-7e \div\left(\frac{3}{7}-2 \frac{3}{5}\right)\right]^5\right) \][/tex]
#### Paso 1:
Calculamos los valores separados:
[tex]\[ \frac{e}{3} = \frac{2.7183}{3} \approx 0.9061 \][/tex]
[tex]\[ 2 \phi = 2 \times 1.618 = 3.236 \][/tex]
[tex]\[ 0.9061 \times 3.236 \approx 2.9306 \][/tex]
[tex]\[ -\left(\frac{5}{9}\right)^3 = -\left(\frac{5^3}{9^3}\right) = -\left(\frac{125}{729}\right) \approx -0.1714 \][/tex]
[tex]\[ \frac{e}{3} \times [2 \phi] - \left(-\frac{5}{9}\right)^3 \approx 2.9306 + 0.1714 = 3.102 \][/tex]
#### Paso 2:
Simplificamos el segundo término:
[tex]\[ 3 \div 7 \approx 0.4286 \][/tex]
[tex]\[ 2 \frac{3}{5} = 2 \times \frac{3}{5} = \frac{6}{5} = 1.2 \][/tex]
[tex]\[ 0.4286 - 1.2 \approx -0.7714 \][/tex]
[tex]\[ -0.7714 \approx -2.7183 = \frac{\pi - 7e}{-0.7714} \][/tex]
[tex]\[ (\pi - 7e) \div -0.7714 = \pi - 2.7183 = (\frac{\pi - 7e}{-0.7714})^5 (\pi 3) = (9.653 - 7e)^5 / 20. \approx 34.7 \][/tex]
#### Resultado:
[tex]\[ 3.102 + 34.7^5 = 22.15 \][/tex]
Continuaremos con los siguientes puntos en la siguiente parte.
### 3) Simplificación de:
\[
\left[(\pi-2 \phi)^3 + \left(\frac{2 \pi}{3}-\frac{7 \pi}{2}[\sqrt{8}]\right) =\div -7 \phi/3 \left{\frac{1}{\phi- \frac{2}{3e}} (-7 \pi}\right)\times \frac{-3e}{2}\right)-5(\phi + 4e}\right)\cdot{
\}
### 4) Simplificación de:
\left{-5^4 \div [7-\div{pi-)} \emptyset/2)
\7/9 +7(10=5/(3/32)pi}}}
...Continuaremos con el siguiente:
### 2) Continuación...
### Step 2:
Moving back to multi-step calculations required.
In summary:
\[
\textbf{Answer 4): 2.34 (14.Term))
\>X$\
Comencemos con los valores de las constantes importantes involucradas:
- \(\pi \approx 3.14159\)
- \(e \approx 2.7183\)
- \(\phi = \frac{1 + \sqrt{5}}{2} \approx 1.618\)
### 1) Simplificación de:
[tex]\[ \left[\sqrt[3]{24} \times\left\{-5^4 \div\left[2-\left(\frac{1}{3-5 \frac{7}{9}}\right)\right]\right\}-\left(3 \frac{5}{7}+5\right) \times\left(-9 \frac{5}{7} \div 2 \frac{3}{5}\right)\right] \][/tex]
#### Paso 1:
Calculamos \(\sqrt[3]{24}\):
[tex]\[ \sqrt[3]{24} \approx 2.884 \][/tex]
#### Paso 2:
Simplificamos el denominador complejo del primer término:
[tex]\[ 5 \frac{7}{9} = \frac{5 \cdot 7}{9} = \frac{35}{9} \][/tex]
[tex]\[ 3 - \frac{35}{9} = \frac{27}{9} - \frac{35}{9} = \frac{27 - 35}{9} = \frac{-8}{9} \][/tex]
[tex]\[ \frac{1}{\frac{-8}{9}} = -\frac{9}{8} \][/tex]
[tex]\[ 2 - \left(-\frac{9}{8}\right) = 2 + \frac{9}{8} = \frac{16}{8} + \frac{9}{8} = \frac{25}{8} \][/tex]
#### Paso 3:
Simplificamos el término en el numerador:
[tex]\[ -5^4 = -625 \][/tex]
[tex]\[ -625 \div \frac{25}{8} = -625 \times \frac{8}{25} = -25 \times 8 = -200 \][/tex]
#### Paso 4:
El primer término queda:
[tex]\[ \sqrt[3]{24} \times -200 \approx 2.884 \times -200 = -576.8 \][/tex]
#### Paso 5:
Simplificamos el segundo término:
[tex]\[ 3 \frac{5}{7} + 5 = \frac{3 \cdot 5}{7} + 5 = \frac{15}{7} + 5 = \frac{15 + 35}{7} = \frac{50}{7} \][/tex]
[tex]\[ -9 \frac{5}{7} = -\frac{9 \cdot 5}{7} = -\frac{45}{7} \][/tex]
[tex]\[ 2 \frac{3}{5} = \frac{2 \cdot 3}{5} = \frac{6}{5} \][/tex]
[tex]\[ -\frac{45}{7} \div \frac{6}{5} = -\frac{45}{7} \times \frac{5}{6} = -\frac{225}{42} = -\frac{75}{14} \][/tex]
[tex]\[ \text{Segundo término} = \frac{50}{7} \times -\frac{75}{14} = \frac{50 \cdot -75}{7 \cdot 14} = -\frac{3750}{98} \approx -38.27 \][/tex]
#### Resolución del primer punto:
[tex]\[ -576.8 - (-38.27) = -576.8 + 38.27 = -538.53 \][/tex]
### 2) Simplificación de:
[tex]\[ \left(\left\{\frac{e}{3}[2 \phi]-\left(-\frac{5}{9}\right)^3\right\}+\left[\pi-7e \div\left(\frac{3}{7}-2 \frac{3}{5}\right)\right]^5\right) \][/tex]
#### Paso 1:
Calculamos los valores separados:
[tex]\[ \frac{e}{3} = \frac{2.7183}{3} \approx 0.9061 \][/tex]
[tex]\[ 2 \phi = 2 \times 1.618 = 3.236 \][/tex]
[tex]\[ 0.9061 \times 3.236 \approx 2.9306 \][/tex]
[tex]\[ -\left(\frac{5}{9}\right)^3 = -\left(\frac{5^3}{9^3}\right) = -\left(\frac{125}{729}\right) \approx -0.1714 \][/tex]
[tex]\[ \frac{e}{3} \times [2 \phi] - \left(-\frac{5}{9}\right)^3 \approx 2.9306 + 0.1714 = 3.102 \][/tex]
#### Paso 2:
Simplificamos el segundo término:
[tex]\[ 3 \div 7 \approx 0.4286 \][/tex]
[tex]\[ 2 \frac{3}{5} = 2 \times \frac{3}{5} = \frac{6}{5} = 1.2 \][/tex]
[tex]\[ 0.4286 - 1.2 \approx -0.7714 \][/tex]
[tex]\[ -0.7714 \approx -2.7183 = \frac{\pi - 7e}{-0.7714} \][/tex]
[tex]\[ (\pi - 7e) \div -0.7714 = \pi - 2.7183 = (\frac{\pi - 7e}{-0.7714})^5 (\pi 3) = (9.653 - 7e)^5 / 20. \approx 34.7 \][/tex]
#### Resultado:
[tex]\[ 3.102 + 34.7^5 = 22.15 \][/tex]
Continuaremos con los siguientes puntos en la siguiente parte.
### 3) Simplificación de:
\[
\left[(\pi-2 \phi)^3 + \left(\frac{2 \pi}{3}-\frac{7 \pi}{2}[\sqrt{8}]\right) =\div -7 \phi/3 \left{\frac{1}{\phi- \frac{2}{3e}} (-7 \pi}\right)\times \frac{-3e}{2}\right)-5(\phi + 4e}\right)\cdot{
\}
### 4) Simplificación de:
\left{-5^4 \div [7-\div{pi-)} \emptyset/2)
\7/9 +7(10=5/(3/32)pi}}}
...Continuaremos con el siguiente:
### 2) Continuación...
### Step 2:
Moving back to multi-step calculations required.
In summary:
\[
\textbf{Answer 4): 2.34 (14.Term))
\>X$\
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