To understand why [tex]\( 7 \cdot 1 = 7 \)[/tex] holds true, we need to identify the correct property of whole-number multiplication that justifies this equality.
1. Commutative Property: This property states that the order of the numbers being multiplied does not affect the product. For any two numbers [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[
a \cdot b = b \cdot a
\][/tex]
For example, [tex]\( 3 \cdot 4 = 4 \cdot 3 \)[/tex]. This property does not specifically explain why multiplying by 1 results in the original number.
2. Associative Property: This property states that the way numbers are grouped in multiplication does not affect the product. For any three numbers [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
(a \cdot b) \cdot c = a \cdot (b \cdot c)
\][/tex]
For example, [tex]\( (2 \cdot 3) \cdot 4 = 2 \cdot (3 \cdot 4) \)[/tex]. This property does not justify the equality [tex]\( 7 \cdot 1 = 7 \)[/tex].
3. Distributive Property: This property relates multiplication and addition. For any three numbers [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
a \cdot (b + c) = (a \cdot b) + (a \cdot c)
\][/tex]
For example, [tex]\( 2 \cdot (3 + 4) = (2 \cdot 3) + (2 \cdot 4) \)[/tex]. This property is not relevant to the equation [tex]\( 7 \cdot 1 = 7 \)[/tex].
4. Multiplicative-Identity Property: This property states that any number multiplied by 1 will result in the original number. For any number [tex]\( a \)[/tex]:
[tex]\[
a \cdot 1 = a
\][/tex]
This property directly explains why [tex]\( 7 \cdot 1 = 7 \)[/tex], because multiplying 7 by 1 leaves it unchanged.
Therefore, the property of whole-number multiplication that justifies the equality [tex]\( 7 \cdot 1 = 7 \)[/tex] is the multiplicative-identity property.
