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Grouping symbols are used to indicate that a particular collection of numbers and meaningful operations are to be grouped together and considered as one number. The grouping symbols commonly used in mathematics are the following:
In a computation in which more than one operation is involved, grouping symbols indicate which operation to perform first. If possible, we perform operations inside grouping symbols first.
If possible, determine the value of each of the following.
$9+(3\cdot 8)$
Since 3 and 8 are within parentheses, they are to be combined first.
$\begin{array}{cc}\hfill 9\phantom{\rule{2px}{0ex}}+\phantom{\rule{2px}{0ex}}(3\phantom{\rule{2px}{0ex}}\cdot \phantom{\rule{2px}{0ex}}8)& =\phantom{\rule{2px}{0ex}}9\phantom{\rule{2px}{0ex}}+\phantom{\rule{2px}{0ex}}\mathrm{24}\hfill \\ & =\phantom{\rule{2px}{0ex}}\mathrm{33}\hfill \end{array}$
Thus,
$9+(3\cdot 8)=\text{33}$
$(\text{10}\xf70)\cdot 6$
Since $\text{10}\xf70$ is undefined, this operation is meaningless, and we attach no value to it. We write, "undefined."
If possible, determine the value of each of the following.
When a set of grouping symbols occurs inside another set of grouping symbols, we perform the operations within the innermost set first.
Determine the value of each of the following.
$2+(8\cdot 3)-(5+6)$
Combine 8 and 3 first, then combine 5 and 6.
$\begin{array}{cc}2+\mathrm{24}-\mathrm{11}\hfill & \text{Now combine left to right.}\\ \mathrm{26}-\mathrm{11}\hfill & \\ \mathrm{15}\hfill & \end{array}$
$\text{10}+[\text{30}-(2\cdot 9)]$
Combine 2 and 9 since they occur in the innermost set of parentheses.
$\begin{array}{cc}\mathrm{10}+[\mathrm{30}-\mathrm{18}]\hfill & \text{Now combine 30 and 18.}\\ \mathrm{10}+\mathrm{12}\hfill & \\ \mathrm{22}\hfill & \end{array}$
Determine the value of each of the following.
Sometimes there are no grouping symbols indicating which operations to perform first. For example, suppose we wish to find the value of $3+5\cdot 2$ . We could do either of two things:
Add 3 and 5, then multiply this sum by 2.
$\begin{array}{cc}\hfill 3+5\cdot 2& =\phantom{\rule{2px}{0ex}}8\phantom{\rule{2px}{0ex}}\cdot \phantom{\rule{2px}{0ex}}2\hfill \\ & =\phantom{\rule{2px}{0ex}}\mathrm{16}\hfill \end{array}$
Multiply 5 and 2, then add 3 to this product.
$\begin{array}{cc}\hfill 3+5\cdot 2& =\phantom{\rule{2px}{0ex}}3\phantom{\rule{2px}{0ex}}+\phantom{\rule{2px}{0ex}}\mathrm{10}\hfill \\ & =\phantom{\rule{2px}{0ex}}\mathrm{13}\hfill \end{array}$
We now have two values for one number. To determine the correct value, we must use the accepted order of operations .
Determine the value of each of the following.
$\begin{array}{cc}\mathrm{21}+3\cdot \mathrm{12}\hfill & \text{Multiply first.}\hfill \\ \mathrm{21}+\mathrm{36}\hfill & \text{Add.}\hfill \\ \mathrm{57}\hfill & \end{array}$
$\begin{array}{cc}(\mathrm{15}-8)+5\cdot (6+4)\mathrm{.}\hfill & \text{Simplify inside parentheses first.}\hfill \\ 7+5\cdot \mathrm{10}\hfill & \text{Multiply.}\hfill \\ 7+\mathrm{50}\hfill & \text{Add.}\hfill \\ \mathrm{57}\hfill & \end{array}$
$\begin{array}{cc}\mathrm{63}-(4+6\cdot 3)+\mathrm{76}-4\hfill & \text{Simplify first within the parenthesis by multiplying, then adding.}\hfill \\ \mathrm{63}-(4+\mathrm{18})+\mathrm{76}-4\hfill & \hfill \\ \mathrm{63}-\mathrm{22}+\mathrm{76}-4\hfill & \text{Now perform the additions and subtractions, moving left to right.}\hfill \\ \mathrm{41}+\mathrm{76}-4\hfill & \text{Add 41 and 76:}\phantom{\rule{8px}{0ex}}\mathrm{41}+\mathrm{76}=\mathrm{117}\mathrm{.}\hfill \\ \mathrm{117}-4\hfill & \text{Subtract 4 from 117:}\phantom{\rule{8px}{0ex}}\mathrm{117}-4=\mathrm{113}\mathrm{.}\hfill \\ \mathrm{113}\hfill & \end{array}$
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