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Multiplying Using the Distributive Property
Multiply , where the outside factor is a variable monomial rather than a numerical constant.
Step 1 — Distribute: Multiply by each term inside the parentheses:
Step 2 — Simplify: For the first term, apply the Product Property for Exponents: . For the second term, :
The result is . When the monomial being distributed is itself a variable, the Product Property for Exponents comes into play — multiplying by produces , not . This distinguishes distributing a variable monomial from distributing a numerical constant, where only arithmetic is needed.
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A payroll specialist is calculating a standard 5 percent bonus for two different departments. The total bonus can be represented by the expression 0.05(S + M), where S is the total salary of the Sales department and M is the total salary of the Marketing department. Which mathematical property justifies rewriting this expression as 0.05S + 0.05M?
A contractor is calculating the total area of two rooms that have the same width 'w' but different lengths 'l1' and 'l2'. The total area is w(l1 + l2). The contractor calculates the area of each room separately as wl1 + wl2 and then adds them. The mathematical rule that justifies this is the ________ property.
A payroll administrator is expanding the expression 40(r + 2) to calculate total weekly pay including a 2 dollar hourly raise. Arrange the steps in the correct order to expand this expression using the Distributive Property.
An HR manager is simplifying payroll formulas for different departments. Match each initial cost expression on the left with its equivalent simplified form on the right using the distributive property.
A facilities manager is calculating the total cost of upgrading n standard workstations and 5 conference rooms, where the cost per upgrade is x dollars. The total cost is represented by the expression x(n + 5). True or False: According to the distributive property, this expression is equivalent to xn + 5.
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A retail manager is calculating the total revenue from selling 'k' holiday gift baskets. Each basket contains a bottle of sparkling cider (c) and a box of chocolates (h). The total revenue is represented by the expression k(c + h). According to the distributive property, which of the following is equivalent to this expression?
A project coordinator is calculating the total labor cost for two separate teams using the expression (Hours_Team_A + Hours_Team_B) × Hourly_Rate. To allocate expenses to different departments, the coordinator rewrites this as (Hours_Team_A × Hourly_Rate) + (Hours_Team_B × Hourly_Rate). Which mathematical property justifies this expansion?
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Learn After
A logistics coordinator is calculating the area of a storage zone using the expression y(y - 2). According to the distributive property, which of the following represents the correct expanded form of this expression?
A facility manager is calculating the area of a rectangular storage unit with a length of 'y' and a width of 'y - 2'. The manager sets up the expression y(y - 2). After applying the distributive property to expand this expression, the resulting expression is ____.
A facility planner is expanding the algebraic expression to calculate the total square footage of a new office suite. Match each part of the distributive process with its correct simplified result to help the planner complete the calculation.
A facility planner is calculating the square footage of a specialized storage zone using the expression . True or False: When the distributive property is applied to expand this expression, the correctly simplified result is .
A facility planner is expanding the area expression to calculate the total square footage of a new storage zone. Arrange the following steps in the correct order to expand this expression using the distributive property.
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A systems administrator is refactoring a configuration script that calculates dynamic server thresholds using the expression . To simplify the script's logic, the administrator needs to expand this expression using the distributive property. Which of the following represents the correct expanded form?
A procurement manager is using the formula to calculate the total cost of a bulk order, where represents the base price and represents the discounted price per unit. To update the company's automated billing system, the manager needs to expand this expression using the distributive property. Which of the following represents the correct expanded form?