2.2: Order of Operations

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Kelly Brooks
Eastern Gateway Community College

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The order of operations represents a mathematical agreement of the order in which calculations should be performed. The order is Grouping Symbols, Exponents, Multiplication and Division as they appear left to right, and Addition and Subtraction as they appear left to right. Traditionally, we use PEMDAS as the acronym to remember the order of operations using Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction. To allow us to perform operations on more problems, let’s consider Grouping Symbols as the first order since it would encompass parentheses, brackets, roots, absolute values, and fraction bars. So, let’s learn the acronym GEMDAS, one way to remember the letters is to remember GEMDAS: Greater Education Makes Doctors and Scholars or Good (Environment) Efforts Minimizes Diseased Animals Swimming

Grouping Symbols-Exponents-Multiplication/Division-Addition/Subtraction

GEMDAS

Example \(\PageIndex{1}\)

Simplify \(5-3(4-6\div (-2))\).

Solution

In this problem, we start with the inner most grouping symbol which would be the parentheses around the -2, however, there is no operation with this set of parentheses, so we move to the other set of parentheses: \((4-6\div (-2))\). Within this set of parentheses, the division must occur before the subtraction, so we computer \(6\div (-2)=-3\), so the parentheses would be

\[(4-(-3))\nonumber \]

Next, we can recall that we subtract integers by adding the opposite, so \((4-(-3))=4+3=7\) continuing, we now have

\[5-3(4-6\div (-2))=5-3(7)\nonumber \]

Next, we perform the multiplication before the subtraction to obtain:

\[5-21=-15\nonumber \]

Example \(\PageIndex{2}\)

Simplify \(2+3\cdot \sqrt{21-5}-8+4^2\).

Solution

In this problem, we start with the radical since it is a grouping symbol:

\[\sqrt{21-5}=\sqrt{16}=4\nonumber \]

So now we have:

\[2+3\cdot 4-8+4^2\nonumber \]

Next, we apply the power or exponent:

\[4^2=16\nonumber \]

So, we now have:

\[2+3\cdot 4-8+16\nonumber \]

Next, we perform multiplication:

\[3\cdot 4=12 \nonumber \]

So, we now have:

\[2+12-8+16\nonumber \]

Finally, we perform addition and subtraction as it appears left to right to obtain:

\[2+12-8+16=14-8+16=6+16=22\nonumber \]