Simplifying Expressions With the Distributive Property Law

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The distributive property is a property (or law) in algebra that dictates how multiplication of a single term operates with two or more terms inside parentheticals and can be used to simplify mathematical expressions that contain sets of parentheses.

Basically, the distributive property of multiplication states that all numbers within the parentheticals must be multiplied individually by the number outside the parentheticals. In other words, the number outside the parentheticals is said to distribute across the numbers inside the parenthesis.

Equations and expressions can be simplified by performing the first step of solving the equation or expression: following the order of operations to multiply the number outside the parentheses by all numbers within the parenthesis then rewriting the equation with the parentheticals removed.

Once this is complete, students can then begin to solve the simplified equation, and depending on how complicated those are; the student may need to further simplify them by moving down the order of operations to multiplication and division then addition and subtraction.

Practicing With Worksheets

Algebra Worksheets

Take a look at the worksheet on the left, which poses a number of mathematical expressions that can be simplified and later solved by first using the distributive property to remove the parentheticals.

In question 1, for instance, the expression -n - 5(-6 - 7n) can be simplified by distributing -5 across the parenthesis and multiplying both -6 and -7n by -5 t get -n + 30 + 35n, which can then be further simplified by combining like values to the expression 30 + 34n.

In each of these expressions, the letter is representative of a range of numbers that could be used in the expression and is most useful when attempting to write mathematical expressions based on word problems.

Another way to get students to arrive at the expression in question 1, for instance, is by saying the negative number minus five times negative six minus seven times a number. 

Using the Distributive Property to Multiply Large Numbers

Algebra Worksheets

Although the worksheet on the left doesn't cover this core concept, students should also understand the importance of the distributive property when multiplying multiple-digit numbers by single-digit numbers (and later multiple-digit numbers).

In this scenario, students would multiply each of the numbers in the multiple-digit number, writing down the ones value of each result in the corresponding place value where the multiplication occurs, carrying any remainders to be added to the next place value.

When multiplying multiple-place-value numbers with others of the same size, students will have to multiply each number in the first by each number in the second, moving over one decimal place and down one row for each number being multiplied in the second.

For example, 1123 multiplied by 3211 could be calculated by first multiplying 1 times 1123 (1123), then moving one decimal value to the left and multiplying 1 by 1123 (11,230) then moving one decimal value to the left and multiplying 2 by 1123 (224,600), then moving one more decimal value to the left and multiply 3 by 1123 (3,369,000), then adding all these numbers together to get 3,605,953.

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Russell, Deb. "Simplifying Expressions With the Distributive Property Law." ThoughtCo, Aug. 27, 2020, Russell, Deb. (2020, August 27). Simplifying Expressions With the Distributive Property Law. Retrieved from Russell, Deb. "Simplifying Expressions With the Distributive Property Law." ThoughtCo. (accessed June 7, 2023).