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Sagot :
Certainly! Let's prove the Commutative and Associative properties for the numbers [tex]\( a = 2,546,789 \)[/tex], [tex]\( b = 7,800,917 \)[/tex], and [tex]\( c = 5,317,920 \)[/tex].
### Commutative Property
Commutative Property of Addition:
The commutative property of addition states that changing the order of the numbers being added does not change the sum. In mathematical terms, [tex]\( a + b = b + a \)[/tex].
Let's verify:
1. Compute [tex]\( a + b \)[/tex]:
[tex]\[ 2,546,789 + 7,800,917 = 10,347,706 \][/tex]
2. Compute [tex]\( b + a \)[/tex]:
[tex]\[ 7,800,917 + 2,546,789 = 10,347,706 \][/tex]
Thus, [tex]\( 2,546,789 + 7,800,917 = 10,347,706 \)[/tex] and [tex]\( 7,800,917 + 2,546,789 = 10,347,706 \)[/tex], confirming the commutative property of addition.
Commutative Property of Multiplication:
The commutative property of multiplication states that changing the order of the numbers being multiplied does not change the product. In mathematical terms, [tex]\( a \times b = b \times a \)[/tex].
Let's verify:
1. Compute [tex]\( a \times b \)[/tex]:
[tex]\[ 2,546,789 \times 7,800,917 = 19,867,289,605,513 \][/tex]
2. Compute [tex]\( b \times a \)[/tex]:
[tex]\[ 7,800,917 \times 2,546,789 = 19,867,289,605,513 \][/tex]
Thus, [tex]\( 2,546,789 \times 7,800,917 = 19,867,289,605,513 \)[/tex] and [tex]\( 7,800,917 \times 2,546,789 = 19,867,289,605,513 \)[/tex], confirming the commutative property of multiplication.
### Associative Property
Associative Property of Addition:
The associative property of addition states that the way in which numbers are grouped when summed does not change the sum. In mathematical terms, [tex]\( (a + b) + c = a + (b + c) \)[/tex].
Let's verify:
1. Compute [tex]\( (a + b) + c \)[/tex]:
[tex]\[ (2,546,789 + 7,800,917) + 5,317,920 = 10,347,706 + 5,317,920 = 15,665,626 \][/tex]
2. Compute [tex]\( a + (b + c) \)[/tex]:
[tex]\[ 2,546,789 + (7,800,917 + 5,317,920) = 2,546,789 + 13,118,837 = 15,665,626 \][/tex]
Thus, [tex]\( (2,546,789 + 7,800,917) + 5,317,920 = 15,665,626 \)[/tex] and [tex]\( 2,546,789 + (7,800,917 + 5,317,920) = 15,665,626 \)[/tex], confirming the associative property of addition.
Associative Property of Multiplication:
The associative property of multiplication states that the way in which numbers are grouped when multiplied does not change the product. In mathematical terms, [tex]\( (a \times b) \times c = a \times (b \times c) \)[/tex].
Let's verify:
1. Compute [tex]\( (a \times b) \times c \)[/tex]:
[tex]\[ (2,546,789 \times 7,800,917) \times 5,317,920 = 19,867,289,605,513 \times 5,317,920 = 105,652,656,738,949,692,960 \][/tex]
2. Compute [tex]\( a \times (b \times c) \)[/tex]:
[tex]\[ 2,546,789 \times (7,800,917 \times 5,317,920) = 2,546,789 \times 41,478,850,649,440 = 105,652,656,738,949,692,960 \][/tex]
Thus, [tex]\( (2,546,789 \times 7,800,917) \times 5,317,920 = 105,652,656,738,949,692,960 \)[/tex] and [tex]\( 2,546,789 \times (7,800,917 \times 5,317,920) = 105,652,656,738,949,692,960 \)[/tex], confirming the associative property of multiplication.
In summary, the given properties have been verified for the numbers [tex]\( a = 2,546,789 \)[/tex], [tex]\( b = 7,800,917 \)[/tex], and [tex]\( c = 5,317,920 \)[/tex]. The Commutative properties of both addition and multiplication hold true, and the Associative properties of both addition and multiplication also hold true.
### Commutative Property
Commutative Property of Addition:
The commutative property of addition states that changing the order of the numbers being added does not change the sum. In mathematical terms, [tex]\( a + b = b + a \)[/tex].
Let's verify:
1. Compute [tex]\( a + b \)[/tex]:
[tex]\[ 2,546,789 + 7,800,917 = 10,347,706 \][/tex]
2. Compute [tex]\( b + a \)[/tex]:
[tex]\[ 7,800,917 + 2,546,789 = 10,347,706 \][/tex]
Thus, [tex]\( 2,546,789 + 7,800,917 = 10,347,706 \)[/tex] and [tex]\( 7,800,917 + 2,546,789 = 10,347,706 \)[/tex], confirming the commutative property of addition.
Commutative Property of Multiplication:
The commutative property of multiplication states that changing the order of the numbers being multiplied does not change the product. In mathematical terms, [tex]\( a \times b = b \times a \)[/tex].
Let's verify:
1. Compute [tex]\( a \times b \)[/tex]:
[tex]\[ 2,546,789 \times 7,800,917 = 19,867,289,605,513 \][/tex]
2. Compute [tex]\( b \times a \)[/tex]:
[tex]\[ 7,800,917 \times 2,546,789 = 19,867,289,605,513 \][/tex]
Thus, [tex]\( 2,546,789 \times 7,800,917 = 19,867,289,605,513 \)[/tex] and [tex]\( 7,800,917 \times 2,546,789 = 19,867,289,605,513 \)[/tex], confirming the commutative property of multiplication.
### Associative Property
Associative Property of Addition:
The associative property of addition states that the way in which numbers are grouped when summed does not change the sum. In mathematical terms, [tex]\( (a + b) + c = a + (b + c) \)[/tex].
Let's verify:
1. Compute [tex]\( (a + b) + c \)[/tex]:
[tex]\[ (2,546,789 + 7,800,917) + 5,317,920 = 10,347,706 + 5,317,920 = 15,665,626 \][/tex]
2. Compute [tex]\( a + (b + c) \)[/tex]:
[tex]\[ 2,546,789 + (7,800,917 + 5,317,920) = 2,546,789 + 13,118,837 = 15,665,626 \][/tex]
Thus, [tex]\( (2,546,789 + 7,800,917) + 5,317,920 = 15,665,626 \)[/tex] and [tex]\( 2,546,789 + (7,800,917 + 5,317,920) = 15,665,626 \)[/tex], confirming the associative property of addition.
Associative Property of Multiplication:
The associative property of multiplication states that the way in which numbers are grouped when multiplied does not change the product. In mathematical terms, [tex]\( (a \times b) \times c = a \times (b \times c) \)[/tex].
Let's verify:
1. Compute [tex]\( (a \times b) \times c \)[/tex]:
[tex]\[ (2,546,789 \times 7,800,917) \times 5,317,920 = 19,867,289,605,513 \times 5,317,920 = 105,652,656,738,949,692,960 \][/tex]
2. Compute [tex]\( a \times (b \times c) \)[/tex]:
[tex]\[ 2,546,789 \times (7,800,917 \times 5,317,920) = 2,546,789 \times 41,478,850,649,440 = 105,652,656,738,949,692,960 \][/tex]
Thus, [tex]\( (2,546,789 \times 7,800,917) \times 5,317,920 = 105,652,656,738,949,692,960 \)[/tex] and [tex]\( 2,546,789 \times (7,800,917 \times 5,317,920) = 105,652,656,738,949,692,960 \)[/tex], confirming the associative property of multiplication.
In summary, the given properties have been verified for the numbers [tex]\( a = 2,546,789 \)[/tex], [tex]\( b = 7,800,917 \)[/tex], and [tex]\( c = 5,317,920 \)[/tex]. The Commutative properties of both addition and multiplication hold true, and the Associative properties of both addition and multiplication also hold true.
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