Answer :
To determine which equation demonstrates the associative property of multiplication, we first need to understand what the associative property is. The associative property of multiplication states that the way in which factors are grouped in a multiplication expression does not affect the product. Mathematically, it can be expressed as:
[tex]\[ (a \times b) \times c = a \times (b \times c) \][/tex]
Now, let's analyze each option to see which one illustrates this property:
Option A:
[tex]\[ 23 \times (4 \times 31) = (4 \times 31) \times 23 \][/tex]
This option reorders the factors but does not regroup them differently on both sides of the equation. Therefore, this does not reflect the associative property.
Option B:
[tex]\[ 46 \times 1 = 46 \][/tex]
This option represents the identity property of multiplication, which states that any number multiplied by 1 remains unchanged. This is not related to the associative property.
Option C:
[tex]\[ 16 \times 31 = 16(30 + 1) \][/tex]
This option utilizes the distributive property of multiplication, which involves distributing the multiplication over addition. It does not exhibit the associative property.
Option D:
[tex]\[ 24 \times (4 \times 51) = (24 \times 4) \times 51 \][/tex]
This option shows that the grouping of factors is reassigned in accordance with the associative property. The left-hand side groups 4 and 51 together first, while the right-hand side groups 24 and 4 together first before multiplying by 51. This conforms to the property:
[tex]\[ (a \times b) \times c = a \times (b \times c) \][/tex]
Hence, the correct equation showing the associative property of multiplication is:
[tex]\[ D. \quad 24 \times (4 \times 51) = (24 \times 4) \times 51 \][/tex]
[tex]\[ (a \times b) \times c = a \times (b \times c) \][/tex]
Now, let's analyze each option to see which one illustrates this property:
Option A:
[tex]\[ 23 \times (4 \times 31) = (4 \times 31) \times 23 \][/tex]
This option reorders the factors but does not regroup them differently on both sides of the equation. Therefore, this does not reflect the associative property.
Option B:
[tex]\[ 46 \times 1 = 46 \][/tex]
This option represents the identity property of multiplication, which states that any number multiplied by 1 remains unchanged. This is not related to the associative property.
Option C:
[tex]\[ 16 \times 31 = 16(30 + 1) \][/tex]
This option utilizes the distributive property of multiplication, which involves distributing the multiplication over addition. It does not exhibit the associative property.
Option D:
[tex]\[ 24 \times (4 \times 51) = (24 \times 4) \times 51 \][/tex]
This option shows that the grouping of factors is reassigned in accordance with the associative property. The left-hand side groups 4 and 51 together first, while the right-hand side groups 24 and 4 together first before multiplying by 51. This conforms to the property:
[tex]\[ (a \times b) \times c = a \times (b \times c) \][/tex]
Hence, the correct equation showing the associative property of multiplication is:
[tex]\[ D. \quad 24 \times (4 \times 51) = (24 \times 4) \times 51 \][/tex]
