# 7.6 Whole numbers: order of operations  (Page 2/2)

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Add 3 and 5, then multiply this sum by 2.

$\begin{array}{}3+5\cdot 2\\ =8\cdot 2\\ =\text{16}\end{array}$

Multiply 5 and 2, then add 3 to this product.

$\begin{array}{}3+5\cdot 2\\ =3+\text{10}\\ =\text{13}\end{array}$

We now have two values for the same expression.

We need a set of rules to guide anyone to one unique value for this kind of expression. Some of these rules are based on convention, while other are forced on up by mathematical logic.

The universally agreed-upon accepted order of operations for evaluating a mathematical expression is as follows:

1. Parentheses (grouping symbols) from the inside out.

By parentheses we mean anything that acts as a grouping symbol, including anything inside symbols such as [  ], {  }, |  |, and $\sqrt{}$ . Any expression in the numerator or denominator of a fraction or in an exponent is also considered grouped, and should be simplified before carrying out further operations.

If there are nested parentheses (parentheses inside parentheses), you work from the innermost parentheses outward.

2. Exponents and other special functions, such as log, sin, cos etc.

3. Multiplications and divisions, from left to right.

4. Additions and subtractions, from left to right.

For example, given: 3 + 15 ÷ 3 + 5 × 2 2+3

The exponent is an implied grouping, so the 2+3 must be evaluated first:

= 3 + 15 ÷ 3 + 5 × 2 5

Now the exponent is carried out:

= 3 +15 ÷ 3 + 5 × 32

Then the multiplication and division, left to right using 15 ÷ 3 = 5 and 5 × 32 = 160:

= 3 + 5 + 160

Finally, the addition, left to right:

= 168

## Examples, order of operation

Determine the value of each of the following.

$\text{21}+3\cdot \text{12}$ .

Multiply first:

= $\text{21}+\text{36}$

= 57

$\left(\text{15}-8\right)+5\left(6+4\right)$ .

Simplify inside parentheses first.

= $7+5\cdot \text{10}$

Multiply.

= $7+\text{50}$

= 57

$\text{63}-\left(4+6\cdot 3\right)+\text{76}-4$ .

Simplify first within the parentheses by multiplying, then adding:

= $\text{63}-\left(4+\text{18}\right)+\text{76}-4$

= $\text{63}-\text{22}+\text{76}-4$

Now perform the additions and subtractions, moving left to right:

= $\text{41}+\text{76}-4$

= $\text{117}-4$

= 113.

$7\cdot 6-{4}^{2}+{1}^{5}$

Evaluate the exponential forms, moving from left to right:

= $7\cdot 6-\text{16}+1$

Multiply 7 · 6:

= $\text{42}-\text{16}+1$

Subtract 16 from 42:

= 26 + 1

= 27.

$6\cdot \left({3}^{2}+{2}^{2}\right)+{4}^{2}$

Evaluate the exponential forms in the parentheses:

= $6\cdot \left(9+4\right)+{4}^{2}$

Add 9 and 4 in the parentheses:

= $6\cdot \left(\text{13}\right)+{4}^{2}$

Evaluate the exponential form ${4}^{2}$ :

= $6\cdot \left(\text{13}\right)+\text{16}$

Multiply 6 and 13:

= $\text{78}+\text{16}$

= 94

$\frac{{6}^{2}+{2}^{2}}{{4}^{2}+6\cdot {2}^{2}}+\frac{{1}^{3}+{8}^{2}}{{\text{10}}^{2}-\text{19}\cdot 5}$ .

= $\frac{\text{36}+4}{\text{16}+6\cdot 4}+\frac{1+\text{64}}{\text{100}-\text{19}\cdot 5}$

= $\frac{\text{36}+4}{\text{16}+\text{24}}+\frac{1+\text{64}}{\text{100}-\text{95}}$

= $\frac{\text{40}}{\text{40}}+\frac{\text{65}}{5}$

= 1+13

= 14

Recall that the bar is a grouping symbol. The fraction $\frac{{6}^{2}+{2}^{2}}{{4}^{2}+6\cdot {2}^{2}}$ is equivalent to $\left({6}^{2}+{2}^{2}\right)÷\left({4}^{2}+6\cdot {2}^{2}\right)$

## Exercises, order of operations

Determine the value of the following:

8 + (32 – 7)

66

(34 + 18 – 2 · 3) + 11

57

8(10) + 4(2 + 3) – (20 + 3 · 15 + 40 – 5)

0

5 · 8 + 42 – 22

52

4(6 2 – 3 3 ) ÷ (4 2 – 4)

9

(8 + 9 · 3) ÷ 7 + 5 · (8 ÷ 4 + 7 + 3 · 5)

125

$\frac{{3}^{3}+{2}^{3}}{{6}^{2}-\text{29}}+5\left(\frac{{8}^{2}+{2}^{4}}{{7}^{2}-{3}^{2}}\right)÷\frac{8\cdot 3+{1}^{8}}{{2}^{3}-3}$

7

## Module review exercises

For the following problems, find each value.

$2+3\cdot \left(8\right)$

48

$1-5\left(8-8\right)$

meaningless

$\text{37}-1\cdot {6}^{2}$

1

$\text{98}÷2÷{7}^{2}$

1

$\left({4}^{2}-2\cdot 4\right)-{2}^{3}$

0

$\text{61}-\text{22}+4\left[3\cdot \left(\text{10}\right)+\text{11}\right]$

203

$\text{121}-4\cdot \left[\left(4\right)\cdot \left(5\right)-\text{12}\right]+\frac{\text{16}}{2}$

97

${2}^{2}\cdot 3+{2}^{3}\left(6-2\right)-\left(3+\text{17}\right)+\text{11}\left(6\right)$

90

$\frac{8\left(6+\text{20}\right)}{8}+\frac{3\left(6+\text{16}\right)}{\text{22}}$

29

$\frac{\left(1+\text{16}\right)-3}{7}+5\left(\text{12}\right)$

62

${1}^{6}+{0}^{8}+{5}^{2}\left(2+8{\right)}^{3}$

25,001

$\frac{5\left({8}^{2}-9\cdot 6\right)}{{2}^{5}-7}+\frac{{7}^{2}-{4}^{2}}{{2}^{4}-5}$

5

$6\left\{2\cdot 8+3\right\}-\left(5\right)\cdot \left(2\right)+\frac{8}{4}+\left(1+8\right)\cdot \left(1+\text{11}\right)$

214

$\text{26}-2\cdot \left\{\frac{6+\text{20}}{\text{13}}\right\}$

22

$\left(\text{10}+5\right)\cdot \left(\text{10}+5\right)-4\cdot \left(\text{60}-4\right)$

1

$\frac{{6}^{2}-1}{{2}^{3}-3}+\frac{{4}^{3}+2\cdot 3}{2\cdot 5}$

14

$\frac{\text{51}}{\text{17}}+7-2\cdot 5\cdot \left(\frac{\text{12}}{3}\right)$

-30

$\left(\text{21}-3\right)\cdot \left(6-1\right)\cdot \left(6\right)+4\left(6+3\right)$

576

$\frac{\left(2+1{\right)}^{3}+{2}^{3}+{1}^{\text{10}}}{{6}^{2}}-\frac{{\text{15}}^{2}-\left[2\cdot 5{\right]}^{2}}{5\cdot {5}^{2}}$

0