Let's match each equation to its corresponding mathematical property step by step.
1. [tex]\((4+5)+2=4+(5+2)\)[/tex]
This equation rearranges the grouping of numbers in addition, without changing their order. This is an example of the associative property of addition.
2. [tex]\(2(2x+4)=4x+8\)[/tex]
This equation demonstrates how multiplication is distributed over addition. Hence, it represents the distributive property.
3. [tex]\((7 \cdot x) \cdot 3=7 \cdot(x \cdot 3)\)[/tex]
This equation rearranges the grouping of numbers in multiplication, without changing their order. This is an example of the associative property of multiplication.
4. [tex]\((8 \cdot x \cdot 2)=(x \cdot 8 \cdot 2)\)[/tex]
This equation changes the order of multiplication but keeps the same group of elements. This is an example of the commutative property of multiplication.
5. [tex]\((7+3)+1=1+(7+3)\)[/tex]
This equation shows adding numbers in different orders that result in the same sum, but changes their places. This is an example of the commutative property of addition.
Summarizing these matches:
1. [tex]\( (4+5)+2=4+(5+2) \)[/tex] - Associative property of addition
2. [tex]\( 2(2x+4)=4x+8 \)[/tex] - Distributive property
3. [tex]\( (7 \cdot x) \cdot 3=7 \cdot (x \cdot 3) \)[/tex] - Associative property of multiplication
4. [tex]\( (8 \cdot x \cdot 2)=(x \cdot 8 \cdot 2) \)[/tex] - Commutative property of multiplication
5. [tex]\( (7+3)+1=1+(7+3) \)[/tex] - Commutative property of addition
