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Given: [tex]a, b \in \mathbb{Z}[/tex].

If in each case [tex]S \in \mathbb{Z}[/tex], simplify the expression for [tex]S[/tex] and indicate the result used at each step. Then, calculate [tex]S[/tex] using the given values of [tex]a[/tex] and [tex]b[/tex].

a) [tex]S = (30a)(-40b)(10a)(-20b)[/tex], where [tex]a = 1[/tex] and [tex]b = -1[/tex].

1. Simplify the expression:
[tex]S = (30a)(-40b)(10a)(-20b)[/tex]

2. Substitute the given values:
[tex]S = (30 \cdot 1)(-40 \cdot -1)(10 \cdot 1)(-20 \cdot -1)[/tex]

3. Calculate the product:
[tex]S = (30)(40)(10)(20)[/tex]

4. Final result:
[tex]S = 30 \cdot 40 \cdot 10 \cdot 20 = 240,000[/tex]

Sagot :

GinnyAnswer
Claro, vamos a resolver el problema paso a paso utilizando los valores dados para \( a \) y \( b \).

Paso 1: Identificar los valores dados:

- \( a = 1 \)
- \( b = -1 \)

Paso 2: Expresar la fórmula general:

La expresión dada es:

[tex]\[ S = (30a)(-40b)(10a)(-20b) \][/tex]

Paso 3: Sustituir los valores dados en la fórmula:

- Sustituyendo \( a = 1 \) y \( b = -1 \):

[tex]\[ S = (30 \cdot 1)(-40 \cdot -1)(10 \cdot 1)(-20 \cdot -1) \][/tex]

Paso 4: Simplificar cada término:

- Simplificar los términos:

[tex]\[ 30 \cdot 1 = 30 \][/tex]
[tex]\[ -40 \cdot -1 = 40 \][/tex] (Recuerda que el producto de dos números negativos es positivo)
[tex]\[ 10 \cdot 1 = 10 \][/tex]
[tex]\[ -20 \cdot -1 = 20 \][/tex]

Paso 5: Multiplicar los términos simplificados:

Combinar los valores obtenidos:

[tex]\[ S = (30)(40)(10)(20) \][/tex]

Paso 6: Realizar la multiplicación de los valores obtenidos:

- Primero, multiplicamos los dos primeros términos:

[tex]\[ 30 \cdot 40 = 1200 \][/tex]

- Luego, multiplicamos el resultado por el siguiente término:

[tex]\[ 1200 \cdot 10 = 12000 \][/tex]

- Finalmente, multiplicamos el resultado por el último término:

[tex]\[ 12000 \cdot 20 = 240000 \][/tex]

Resultado final:

[tex]\[ S = 240000 \][/tex]

Por lo tanto, el valor de [tex]\( S \)[/tex] es [tex]\( 240000 \)[/tex].
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