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When tackling algebraic equations, breaking them into manageable pieces can dramatically simplify the problem. The distributive property is the mathematical rule that lets you do just that—rearrange and combine terms to make complex expressions easier to handle.
The distributive property links multiplication and addition: multiplying a number by a sum is the same as multiplying it by each addend separately and then adding the results.
Formally:
\(a × (x + y) = ax + ay\)
For example:
\(3 × (4 + 5) = 3 × 4 + 3 × 5\)
Breaking down an expression into smaller parts not only speeds up calculation but also improves conceptual understanding.
Students often encounter the distributive property when learning to multiply larger numbers that require carrying. By rounding a factor to the nearest multiple of ten, you can split the problem into simpler calculations.
Example: Solve \(36 × 4\).
Rewrite it as \(4 × (30 + 6)\) and apply the property:
(\(4 × 30\)) + (\(4 × 6\)) = 120 + 24 = 144
The same principle works in algebraic expressions. If you see an equation like \(a × (b + c)\), distribute the outer factor over the terms inside the parentheses:
\(a × (b + c) = (ab) + (ac)\)
Example:
\(3 × (2 + 4) = (3 × 2) + (3 × 4) = 6 + 12 = 18\)
Recombining terms can also simplify work. For instance:
\(16 × 6 + 16 × 4 = 16 × (6 + 4) = 16 × 10 = 160\)
Try these to reinforce your understanding:
Using the distributive property, you can solve each quickly and confidently.
