# Breaking Down the Distributive Property and Associative Property ## Distributive property:

The distributive property is the process of distributing a group of numbers across other groups.

The distributive property can be demonstrated on an algebraic equation such as (2x + 3)(4y – 7). In this example, 8 is distributed to both 2 and 4 in the parentheses. Thus, it results in 16x and 32y – 21.

This principle also applies to parenthesis with more than two terms inside them. For example: (2x + 3)(4 + 5). In this case, there are only three terms inside the parenthesis but they contain four variables or constants, so 8 goes to each term resulting in 4x and 20 being grouped at the end of the equation.

It can also be used to simplify problems that consist of a series of additions and subtractions with the same variable inside them. For example: (5x – 2) + (3x – 10). In this case, 5x is combined with the second term of 3x, leaving one term which can then be simplified by adding 2 back. Alternatively, if a problem consists of two or more terms where only subtraction is present, distribution should not take place because it would result in division by zero.

## Associative property:

The associative property allows for easier calculation involving parenthesis even when there are more than three terms within them. It states that grouping numbers does not matter and can be applied to expressions (4 + 5)(6 + 7). In this case, the numbers 4 and 6 would be distributed to 5 and 7 resulting in 20 and 84 being grouped. The 4 and 6 would also be grouped, leaving two groups of 8.

These principles can simplify problems involving additions and subtractions with more than one term inside them even when there are multiple parentheses present, but it is essential not to forget about an associative property when using the distributive property because it could result in a different answer or even a wrong answer entirely.

## Uses:

• It is used to simplify problems with multiple additions and subtractions even if there are more than two terms present within them.
• Distributes numbers to other groups in an equation [e.g. (2x + 3)(4y – 7)]
• It is used to combine terms inside one set of parentheses even if there are more than two terms present within them.
• It is not applicable when the number of terms inside parenthesis is increased from three to four or more because this would result in division by zero

### A real-life example of a distributive property:

• putting marbles into equally sized groups to keep them even.

### A real-life example of an associative property:

How much it costs to purchase 1, 2, and 3 items at different stores. For example, this works for buying candy bars at two different stores that are each priced differently. However, it would not work for purchasing three candy bars at one store because the third bar cannot be bought anywhere else. Thus if Quinn bought 1 candy bar at one store for Rs 1 and another candy bar at a second store costing Rs 2 together, it would cost her Rs 3. If she chose to buy all three of the items in such a manner, then it would cost her (1 + 2) = 3 dollars; however, if she went to the first store and bought an item but then went to the second store, it would cost her \$1 + 2 = 3 dollars. Thus, this is why associativity holds. This occurs because how items are grouped or combined does not matter.

So these were some primary information on distributive property and associative property. To find more about these, one can explore the Cuemath website.