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Claro, veamos cómo podemos asociar los factores de al menos dos formas distintas en cada caso y calcular el producto. Procederé de manera detallada:
### a. [tex]\(3 \times 5 \times 4 \times 6\)[/tex]
1. Primera forma: [tex]\((3 \times 4) \times (5 \times 6)\)[/tex]
- Calculamos [tex]\(3 \times 4 = 12\)[/tex]
- Calculamos [tex]\(5 \times 6 = 30\)[/tex]
- Luego, [tex]\(12 \times 30 = 360\)[/tex]
2. Segunda forma: [tex]\((3 \times 6) \times (5 \times 4)\)[/tex]
- Calculamos [tex]\(3 \times 6 = 18\)[/tex]
- Calculamos [tex]\(5 \times 4 = 20\)[/tex]
- Luego, [tex]\(18 \times 20 = 360\)[/tex]
El resultado es [tex]\(360\)[/tex].
---
### b. [tex]\(21 \times 4 \times 6 \times 12\)[/tex]
1. Primera forma: [tex]\((21 \times 4) \times (6 \times 12)\)[/tex]
- Calculamos [tex]\(21 \times 4 = 84\)[/tex]
- Calculamos [tex]\(6 \times 12 = 72\)[/tex]
- Luego, [tex]\(84 \times 72 = 6048\)[/tex]
2. Segunda forma: [tex]\((21 \times 6) \times (4 \times 12)\)[/tex]
- Calculamos [tex]\(21 \times 6 = 126\)[/tex]
- Calculamos [tex]\(4 \times 12 = 48\)[/tex]
- Luego, [tex]\(126 \times 48 = 6048\)[/tex]
El resultado es [tex]\(6048\)[/tex].
---
### c. [tex]\(23 \times 2 \times 12 \times 4\)[/tex]
1. Primera forma: [tex]\((23 \times 2) \times (12 \times 4)\)[/tex]
- Calculamos [tex]\(23 \times 2 = 46\)[/tex]
- Calculamos [tex]\(12 \times 4 = 48\)[/tex]
- Luego, [tex]\(46 \times 48 = 2208\)[/tex]
2. Segunda forma: [tex]\((23 \times 12) \times (2 \times 4)\)[/tex]
- Calculamos [tex]\(23 \times 12 = 276\)[/tex]
- Calculamos [tex]\(2 \times 4 = 8\)[/tex]
- Luego, [tex]\(276 \times 8 = 2208\)[/tex]
El resultado es [tex]\(2208\)[/tex].
---
### d. [tex]\(31 \times 12 \times 12 \times 5 \times 63\)[/tex]
1. Primera forma: [tex]\((31 \times 12) \times (12 \times 5 \times 63)\)[/tex]
- Calculamos [tex]\(31 \times 12 = 372\)[/tex]
- Calculamos [tex]\(12 \times 5 = 60\)[/tex]
- Calculamos [tex]\(60 \times 63 = 3780\)[/tex]
- Luego, [tex]\(372 \times 3780 = 1406160\)[/tex]
2. Segunda forma: [tex]\((31 \times 12 \times 12) \times (5 \times 63)\)[/tex]
- Calculamos [tex]\(31 \times 12 = 372\)[/tex]
- Calculamos nuevamente [tex]\(372 \times 12 = 4464\)[/tex]
- Calculamos luego [tex]\(5 \times 63 = 315\)[/tex]
- Luego, [tex]\(4464 \times 315 = 1406160\)[/tex]
El resultado es [tex]\(1406160\)[/tex].
---
### e. [tex]\(4 \times 5 \times 6 \times 17 \times 9\)[/tex]
1. Primera forma: [tex]\((4 \times 5) \times (6 \times 17 \times 9)\)[/tex]
- Calculamos [tex]\(4 \times 5 = 20\)[/tex]
- Calculamos [tex]\(6 \times 17 = 102\)[/tex]
- Calculamos luego [tex]\(102 \times 9 = 918\)[/tex]
- Luego, [tex]\(20 \times 918 = 18360\)[/tex]
2. Segunda forma: [tex]\((4 \times 6 \times 17) \times (5 \times 9)\)[/tex]
- Calculamos [tex]\(4 \times 6 = 24\)[/tex]
- Calculamos luego [tex]\(24 \times 17 = 408\)[/tex]
- Calculamos luego [tex]\(5 \times 9 = 45\)[/tex]
- Luego, [tex]\(408 \times 45 = 18360\)[/tex]
El resultado es [tex]\(18360\)[/tex].
---
### f. [tex]\(3 \times 7 \times 15 \times 12\)[/tex]
1. Primera forma: [tex]\((3 \times 7) \times (15 \times 12)\)[/tex]
- Calculamos [tex]\(3 \times 7 = 21\)[/tex]
- Calculamos [tex]\(15 \times 12 = 180\)[/tex]
- Luego, [tex]\(21 \times 180 = 3780\)[/tex]
2. Segunda forma: [tex]\((3 \times 15) \times (7 \times 12)\)[/tex]
- Calculamos [tex]\(3 \times 15 = 45\)[/tex]
- Calculamos [tex]\(7 \times 12 = 84\)[/tex]
- Luego, [tex]\(45 \times 84 = 3780\)[/tex]
El resultado es [tex]\(3780\)[/tex].
---
En resumen, los productos calculados para cada caso son:
a. [tex]\(360\)[/tex]
b. [tex]\(6048\)[/tex]
c. [tex]\(2208\)[/tex]
d. [tex]\(1406160\)[/tex]
e. [tex]\(18360\)[/tex]
f. [tex]\(3780\)[/tex]
### a. [tex]\(3 \times 5 \times 4 \times 6\)[/tex]
1. Primera forma: [tex]\((3 \times 4) \times (5 \times 6)\)[/tex]
- Calculamos [tex]\(3 \times 4 = 12\)[/tex]
- Calculamos [tex]\(5 \times 6 = 30\)[/tex]
- Luego, [tex]\(12 \times 30 = 360\)[/tex]
2. Segunda forma: [tex]\((3 \times 6) \times (5 \times 4)\)[/tex]
- Calculamos [tex]\(3 \times 6 = 18\)[/tex]
- Calculamos [tex]\(5 \times 4 = 20\)[/tex]
- Luego, [tex]\(18 \times 20 = 360\)[/tex]
El resultado es [tex]\(360\)[/tex].
---
### b. [tex]\(21 \times 4 \times 6 \times 12\)[/tex]
1. Primera forma: [tex]\((21 \times 4) \times (6 \times 12)\)[/tex]
- Calculamos [tex]\(21 \times 4 = 84\)[/tex]
- Calculamos [tex]\(6 \times 12 = 72\)[/tex]
- Luego, [tex]\(84 \times 72 = 6048\)[/tex]
2. Segunda forma: [tex]\((21 \times 6) \times (4 \times 12)\)[/tex]
- Calculamos [tex]\(21 \times 6 = 126\)[/tex]
- Calculamos [tex]\(4 \times 12 = 48\)[/tex]
- Luego, [tex]\(126 \times 48 = 6048\)[/tex]
El resultado es [tex]\(6048\)[/tex].
---
### c. [tex]\(23 \times 2 \times 12 \times 4\)[/tex]
1. Primera forma: [tex]\((23 \times 2) \times (12 \times 4)\)[/tex]
- Calculamos [tex]\(23 \times 2 = 46\)[/tex]
- Calculamos [tex]\(12 \times 4 = 48\)[/tex]
- Luego, [tex]\(46 \times 48 = 2208\)[/tex]
2. Segunda forma: [tex]\((23 \times 12) \times (2 \times 4)\)[/tex]
- Calculamos [tex]\(23 \times 12 = 276\)[/tex]
- Calculamos [tex]\(2 \times 4 = 8\)[/tex]
- Luego, [tex]\(276 \times 8 = 2208\)[/tex]
El resultado es [tex]\(2208\)[/tex].
---
### d. [tex]\(31 \times 12 \times 12 \times 5 \times 63\)[/tex]
1. Primera forma: [tex]\((31 \times 12) \times (12 \times 5 \times 63)\)[/tex]
- Calculamos [tex]\(31 \times 12 = 372\)[/tex]
- Calculamos [tex]\(12 \times 5 = 60\)[/tex]
- Calculamos [tex]\(60 \times 63 = 3780\)[/tex]
- Luego, [tex]\(372 \times 3780 = 1406160\)[/tex]
2. Segunda forma: [tex]\((31 \times 12 \times 12) \times (5 \times 63)\)[/tex]
- Calculamos [tex]\(31 \times 12 = 372\)[/tex]
- Calculamos nuevamente [tex]\(372 \times 12 = 4464\)[/tex]
- Calculamos luego [tex]\(5 \times 63 = 315\)[/tex]
- Luego, [tex]\(4464 \times 315 = 1406160\)[/tex]
El resultado es [tex]\(1406160\)[/tex].
---
### e. [tex]\(4 \times 5 \times 6 \times 17 \times 9\)[/tex]
1. Primera forma: [tex]\((4 \times 5) \times (6 \times 17 \times 9)\)[/tex]
- Calculamos [tex]\(4 \times 5 = 20\)[/tex]
- Calculamos [tex]\(6 \times 17 = 102\)[/tex]
- Calculamos luego [tex]\(102 \times 9 = 918\)[/tex]
- Luego, [tex]\(20 \times 918 = 18360\)[/tex]
2. Segunda forma: [tex]\((4 \times 6 \times 17) \times (5 \times 9)\)[/tex]
- Calculamos [tex]\(4 \times 6 = 24\)[/tex]
- Calculamos luego [tex]\(24 \times 17 = 408\)[/tex]
- Calculamos luego [tex]\(5 \times 9 = 45\)[/tex]
- Luego, [tex]\(408 \times 45 = 18360\)[/tex]
El resultado es [tex]\(18360\)[/tex].
---
### f. [tex]\(3 \times 7 \times 15 \times 12\)[/tex]
1. Primera forma: [tex]\((3 \times 7) \times (15 \times 12)\)[/tex]
- Calculamos [tex]\(3 \times 7 = 21\)[/tex]
- Calculamos [tex]\(15 \times 12 = 180\)[/tex]
- Luego, [tex]\(21 \times 180 = 3780\)[/tex]
2. Segunda forma: [tex]\((3 \times 15) \times (7 \times 12)\)[/tex]
- Calculamos [tex]\(3 \times 15 = 45\)[/tex]
- Calculamos [tex]\(7 \times 12 = 84\)[/tex]
- Luego, [tex]\(45 \times 84 = 3780\)[/tex]
El resultado es [tex]\(3780\)[/tex].
---
En resumen, los productos calculados para cada caso son:
a. [tex]\(360\)[/tex]
b. [tex]\(6048\)[/tex]
c. [tex]\(2208\)[/tex]
d. [tex]\(1406160\)[/tex]
e. [tex]\(18360\)[/tex]
f. [tex]\(3780\)[/tex]
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