Identify the property illustrated in each statement. Assume all variables represent real numbers. (5x) • (1/5x) = 5 ( x • 1/x )

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Observe the given expression: $(5x) \left( \frac{1}{x} \right) = 5 \left( x \cdot \frac{1}{x} \right)$.

Recognize that the expression shows how multiplication distributes over another operation, specifically how a product can be separated or grouped.

Recall the Distributive Property, which states that for all real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$. However, here we are dealing with multiplication and division, so consider how multiplication interacts with division.

Notice that multiplying \$5x$ by $\frac{1}{x}$ can be rewritten as $5$ times the product of $x$ and $\frac{1}{x}$, which is exactly what the right side shows: $5 \left( x \cdot \frac{1}{x} \right)$.

Conclude that this illustrates the Associative Property of Multiplication, which allows us to regroup factors without changing the product, i.e., $(ab)c = a(bc)$.

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Distributive Property of Multiplication over Division

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Real numbers follow specific algebraic rules such as commutativity, associativity, and distributivity. Recognizing these properties helps in manipulating and simplifying expressions involving variables and constants, as seen in the problem.

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