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

Verified step by step guidance

1. Observe the given expression: \((5x) \left( \frac{1}{x} \right) = 5 \left( x \cdot \frac{1}{x} \right)\).
2. Recognize that the expression shows how multiplication distributes over another operation, specifically how a product can be separated or grouped.
3. 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.
4. 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: \(5 \left( x \cdot \frac{1}{x} \right)\).
5. Conclude that this illustrates the Associative Property of Multiplication, which allows us to regroup factors without changing the product, i.e., \((ab)c = a(bc)\).

Key Concepts

Distributive Property of Multiplication over Division

This property states that multiplying a product by a fraction is equivalent to multiplying each factor by the fraction separately. In the given expression, (5x) * (1/x) = 5 * (x * 1/x), the multiplication distributes over the division, simplifying the expression.

Simplification of Fractions

Simplifying fractions involves reducing expressions by canceling common factors in the numerator and denominator. Here, x in the numerator and denominator cancels out, which is essential to understand how the expression simplifies to 5.

Properties of Real Numbers

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.