Order of Operations

Operations in an expression are performed in a specific order: operations within grouping symbols, exponents, multiplication and division from left to right, and addition and subtraction from left to right.

Operations performed on expressions include addition, subtraction, multiplication, division, exponents, and roots. The order of operations is a set of rules indicating which calculations to perform first to simplify a mathematical expression. Following the order of operations ensures that different people simplifying the same expression will get the same result.

The first step in the order of operations is to simplify within any grouping symbols. Grouping symbols, such as parentheses $(\; )$ , brackets $[ \;]$ , and braces $\lbrace\;\rbrace$ , that are used to separate a part of an expression. Some operation symbols, such as fraction bars and radicals, may also function as grouping symbols. For nested grouping symbols ${ [ ( \;) ] }$ , start with the innermost set, and work to the outside.

Then simplify any exponents, then all multiplication and division from left to right, and then addition and subtraction from left to right.

Memory aids, such as GEMDAS and PEMDAS, are sometimes used to help remember the order of operations. However, it is important to remember that multiplication and division are performed together from left to right; do not perform all multiplication and then all division. Similarly, addition and subtraction are performed together from left to right.

When evaluating an expression with multiple operations, the steps must be performed in the correct order so that the answer is always the same.

Evaluating Expressions with the Order of Operations

Simplify the expression:

\left[1+2(4\cdot3-9)\right]-\sqrt{10-3+2}+\frac{2^3}{11-7}

Simplify the expressions within grouping symbols. Start with the innermost grouping symbol and then work outward.

Simplify the expression in parentheses first. Perform the multiplication and then the subtraction.

\begin{gathered}\left[1+2{\color{#c42126}(4\cdot3-9)}\right]-\sqrt{10-3+2}+\frac{2^3}{11-7}\\\left[1+2{\color{#c42126}(12-9)}\right]-\sqrt{10-3+2}+\frac{2^3}{11-7}\\\left[1+2{\color{#c42126}(3)}\right]-\sqrt{10-3+2}+\frac{2^3}{11-7}\end{gathered}

Next, simplify the expression within the brackets. Perform the multiplication and then the addition.

\begin{gathered}\left[1+{\color{#c42126}6}\right]-\sqrt{10-3+2}+\frac{2^3}{11-7}\\ {\color{#c42126}7}-\sqrt{10-3+2}+\frac{2^3}{11-7}\end{gathered}

Simplify the radicand. Calculate from left to right by first performing the subtraction and then the addition.

\begin{gathered}7-\sqrt{\color{#c42126}{10-3+2}}+\frac{2^3}{11-7}\\7-\sqrt{{\color{#c42126}7+2}}+\frac{2^3}{11-7}\\7-\sqrt{{\color{#c42126}9}}+\frac{2^3}{11-7}\end{gathered}

Simplify the denominator by performing the subtraction.

\begin{gathered}7-\sqrt{9}+\frac{2^3}{{\color{#c42126}11-7}}\\7-\sqrt{9}+\frac{2^3}{{\color{#c42126}4}}\end{gathered}

Simplify any exponents or radicals. Start with the radical in the expression.

\begin{gathered}7-{\color{#c42126}\sqrt{9}}+\frac{2^3}{4}\\7-{\color{#c42126}3}+\frac{2^3}{4}\end{gathered}

Simplify the exponent in the expression.

\begin{gathered}7-3+\frac{{\color{#c42126}2^3}}{4}\\7-3+\frac{{\color{#c42126}8}}{4}\end{gathered}

Perform multiplication and division from left to right.

The fraction

\frac{8}{4}

represents division. So, divide:

\begin{gathered}7-3+{\color{#c42126}\frac{8}{4}}\\7-3+{\color{#c42126}2}\end{gathered}

Perform addition and subtraction.

Calculate from left to right by first performing the subtraction and then the addition.

\begin{gathered}{\color{#c42126}7-3}+2\\{\color{#c42126}4}+2\\{\color{#c42126}6}\end{gathered}

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Order of Operations