The Associative Property states that the way in which numbers are grouped in an addition or multiplication operation does not affect the result. This property applies to both addition and multiplication.
Let's analyze each of the given equations to see if they demonstrate the Associative Property.
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\(3x\cdot (4x+5x)=(3x+4x)\cdot 5\)
This equation is actually applying the Distributive Property rather than the Associative Property. It involves addition in the parentheses, which does not show associativity in multiplication.
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\(3x\cdot (4x\cdot 5)=3x(4x+5)\)
This again combines multiplication and addition, and does not clearly show the associative grouping for multiplication.
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\((3x+4x)\cdot 5=3x+(4x\cdot 5)\)
This expression is using addition and does not involve any associative grouping related to multiplication.
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\(3x\cdot (4x\cdot 5)=(3x\cdot 4x)\cdot 5\)
In this equation, both sides involve multiple products, and they demonstrate the Associative Property of multiplication. The left side groups \(4x\cdot 5\) first and then multiplies by \(3x\), while the right side groups \(3x\cdot 4x\) first and then multiplies by \(5\). Both represent the same result, showing that multiplication is associative.
Thus, the equation that demonstrates the Associative Property is:
\[ \boxed{3x\cdot (4x\cdot 5)=(3x\cdot 4x)\cdot 5} \]
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