When to use the distributive property vs the order of operations?

Question:

When to use the distributive property vs the order of operations?

The Order of Operations and the Distributive Property:

In math, we have a specific order of operations that we use to simplify expressions, and it follows the acronym PEMDAS, standing for parentheses, exponents, multiplication, division, addition, subtraction. The distributive property states that we can multiply a mathematical expression, A, by a sum of mathematical expressions, B + C, using the formula A(B + C) = AC + BC. The distributive property shows up in the order of operations in certain instances.

Answer and Explanation:

Technically speaking, when simplifying mathematical expressions, the distributive property will be used in the correct order as the order of operations. It will show up at the multiplication step. That is, we use the distributive property to simplify an expression when the expression contains parentheses containing a sum that cannot be simplified and is not raised to an exponent.

When a mathematical expression contains parentheses, the order of operations says to first simplify whatever is in parentheses, and then move onto the next operation, which is exponents. Therefore, we always simplify what is in parentheses as much as possible, and then move onto exponents. The distributive property would come in at the multiplication step of order of operations if the parentheses at this step contain a sum that cannot be simplified any further.

For example, suppose we want to simplify the following expression.

  • 4(x + 2 - 1)2 - 7

We would go to our order of operations. The first thing we would do is simplify what is in the parentheses by subtracting 1 from 2.

  • 4(x + 1)2 - 7

Next, we would apply the exponent, so we would square x + 1, or multiply (x + 1)(x + 1).

  • 4(x2 + 2x + 1) - 7

Notice the expression within the parentheses is a sum that cannot be simplified any further, and it is not raised to an exponent. The next step in the order of operations is multiplication, and this would be where our distributive property comes into play, because we would multiply 4 times x2 + 2x + 1 using the distributive property.

  • 4x2 + 8x + 4 - 7

After this, we would move onto division, which there isn't any of. Then onto addition, and there are no like terms being combined with addition. Lastly, we would perform any subtraction of like terms, so we would subtract 7 from 4.

  • 4x2 + 8x - 3

We get that 4(x + 2 - 1)2 - 7 simplifies to 4x2 + 8x - 3, and we see that the distributive property is applied within the order of operations when we have a sum within parentheses that cannot be simplified and is not raised to an exponent.


Learn more about this topic:

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The Distributive Property and Algebraic Expressions

from ELM: CSU Math Study Guide

Chapter 6 / Lesson 5
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