Question

current skill $\\$\
determine which property of multiplication is being demonstrated by each equation listed below.\
select the correct option for each equation.\

equation	commutative	associative	identity	\
---	---	---	---	\
$67 \times (21 \times 57) = (67 \times 21) \times 57$	$\square$	$\square$	$\square$	\
$72 \times (43 \times 31) = (72 \times 43) \times 31$	$\square$	$\square$	$\square$	\
$85 \times 7 = 7 \times 85$	$\square$	$\square$	$\square$

Explanation:

To solve this, we recall the properties of multiplication:

1. For \( 67 \times (21 \times 57) = (67 \times 21) \times 57 \)

• Associative Property: \( (a \times b) \times c = a \times (b \times c) \) (grouping of factors changes, product remains same).
• Here, the grouping of \( 67, 21, 57 \) changes (from \( 21 \times 57 \) grouped first to \( 67 \times 21 \) grouped first), so this is the Associative Property.

2. For \( 72 \times (43 \times 31) = (72 \times 43) \times 31 \)

• Using the same logic as above, the grouping of \( 72, 43, 31 \) changes. This matches the Associative Property (\( (a \times b) \times c = a \times (b \times c) \)).

3. For \( 85 \times 7 = 7 \times 85 \)

• Commutative Property: \( a \times b = b \times a \) (order of factors changes, product remains same).
• Here, \( 85 \) and \( 7 \) swap places, so this is the Commutative Property.

Final Selections:

• \( 67 \times (21 \times 57) = (67 \times 21) \times 57 \): Check the “Associative” column.
• \( 72 \times (43 \times 31) = (72 \times 43) \times 31 \): Check the “Associative” column.
• \( 85 \times 7 = 7 \times 85 \): Check the “Commutative” column.

Answer:

To solve this, we recall the properties of multiplication:

1. For \( 67 \times (21 \times 57) = (67 \times 21) \times 57 \)

• Associative Property: \( (a \times b) \times c = a \times (b \times c) \) (grouping of factors changes, product remains same).
• Here, the grouping of \( 67, 21, 57 \) changes (from \( 21 \times 57 \) grouped first to \( 67 \times 21 \) grouped first), so this is the Associative Property.

2. For \( 72 \times (43 \times 31) = (72 \times 43) \times 31 \)

• Using the same logic as above, the grouping of \( 72, 43, 31 \) changes. This matches the Associative Property (\( (a \times b) \times c = a \times (b \times c) \)).

3. For \( 85 \times 7 = 7 \times 85 \)

• Commutative Property: \( a \times b = b \times a \) (order of factors changes, product remains same).
• Here, \( 85 \) and \( 7 \) swap places, so this is the Commutative Property.

Final Selections:

• \( 67 \times (21 \times 57) = (67 \times 21) \times 57 \): Check the “Associative” column.
• \( 72 \times (43 \times 31) = (72 \times 43) \times 31 \): Check the “Associative” column.
• \( 85 \times 7 = 7 \times 85 \): Check the “Commutative” column.