Order of Operations and Simplifying

Section 1.3 Order of Operations and Simplifying

Whenever we have a mathematical expression with multiple things you need to do, we all have to agree on the default order in which to do them. Think of it as part of the language of mathematics. We write down these expressions to communicate something, but that communication is only successful if we view the symbols the same way. The agreed order is often called the Order of Operations.

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Fact 1.3.0.12. Order of Operations.

Whenever you have multiple things to do in a mathematical expression, you must do them in the following order:

Anything inside of a Group (e.g. inside parentheses, top or bottom of a fraction, inside a root or radical, etc.)

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Exponents

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Multiply/Divide

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Add/Subtract

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Sometimes, we use the following abbreviation to help us remember this order: GEMDAS.

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Example 1.3.0.13.

Suppose we want to simplify the following expression:

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\begin{equation*} \frac{8+\sqrt{5(2)+6}}{3^2-5} \end{equation*}

The first thing we need to do is anything inside of a Group. Since we have a fraction, we will deal with the top and bottom of the fraction separately, then divide them. Since the top of a fraction has a root inside of it, we will start with that. Within each group, we follow the order of operations, then move our way out to the next group, following the order of operations again.

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\begin{align*} \frac{8+\sqrt{{\color{red}{5(2)}}+6}}{3^2-5}\amp= \frac{8+\sqrt{{\color{red}{10}}+6}}{3^2-5}\\ \frac{8+\sqrt{\color{red}{10+6}}}{3^2-5}\amp= \frac{8+\sqrt{\color{red}{16}}}{3^2-5}\\ \frac{8+{\color{red}{\sqrt{16}}}}{3^2-5}\amp= \frac{8+{\color{red}{4}}}{3^2-5}\\ \frac{\color{red}{8+4}}{3^2-5}\amp= \frac{\color{red}{12}}{3^2-5}\\ \frac{12}{{\color{red}{3^2}}-5}\amp= \frac{12}{{\color{red}{9}}-5}\\ \frac{12}{\color{red}{9-5}}\amp=\frac{12}{\color{red}{4}}\\ \frac{12}{4}\amp=3 \end{align*}

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