# 3.2 Grouping symbols and the order of operations  (Page 2/2)

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$\begin{array}{cc}7\cdot 6-{4}^{2}+{1}^{5}\hfill & \text{Evaluate the exponential forms, moving left to right.}\hfill \\ 7\cdot 6-\text{16}+1\hfill & \text{Multiply 7 and 6:}\phantom{\rule{8px}{0ex}}7\cdot 6=\text{42}\hfill \\ \text{42}-\text{16}+1\hfill & \text{Subtract 16 from 42:}\phantom{\rule{8px}{0ex}}\text{42}-\text{16}=\text{26}\hfill \\ \text{26}+1\hfill & \text{Add 26 and 1:}\phantom{\rule{8px}{0ex}}\text{26}+1=\text{27}\hfill \\ \mathrm{27}\hfill & \end{array}$

$\begin{array}{cc}\frac{{6}^{2}+{2}^{2}}{{4}^{2}+6\cdot {2}^{2}}+\frac{{1}^{3}+{8}^{2}}{{\text{10}}^{2}-\text{19}\cdot 5}\hfill & \begin{array}{c}\text{Recall that the bar is a grouping symbol.}\hfill \\ \text{The fraction}\frac{{6}^{2}+{2}^{2}}{{4}^{2}+6\cdot {2}^{2}}\text{is equivalent to}\left({6}^{2}+{2}^{2}\right)÷\left({4}^{2}+6\cdot {2}^{2}\right)\hfill \end{array}\hfill \\ \frac{\text{36}+4}{\text{16}+6\cdot 4}+\frac{1+\text{64}}{\text{100}-\text{19}\cdot 5}\hfill & \hfill \\ \frac{\text{36}+4}{\text{16}+\text{24}}+\frac{1+\text{64}}{\text{100}-\text{95}}\hfill & \hfill \\ \frac{\text{40}}{\text{40}}+\frac{\text{65}}{5}\hfill & \hfill \\ 1+\text{13}\hfill & \hfill \\ \text{14}\hfill & \end{array}$

## Practice set c

Determine the value of each of the following.

$8+\left(\text{32}-7\right)$

33

$\left(\text{34}+\text{18}-2\cdot 3\right)+\text{11}$

57

$8\left(\text{10}\right)+4\left(2+3\right)-\left(\text{20}+3\cdot \text{15}+\text{40}-5\right)$

0

$5\cdot 8+{4}^{2}-{2}^{2}$

52

$4\left({6}^{2}-{3}^{3}\right)÷\left({4}^{2}-4\right)$

3

$\left(8+9\cdot 3\right)÷7+5\cdot \left(8÷4+7+3\cdot 5\right)$

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

## Calculators

Using a calculator is helpful for simplifying computations that involve large num­bers.

## Sample set d

Use a calculator to determine each value.

$9,\text{842}+\text{56}\cdot \text{85}$

 Key Display Reads Perform the multiplication first. Type 56 56 Press × 56 Type 85 85 Now perform the addition. Press + 4760 Type 9842 9842 Press = 14602

$\text{42}\left(\text{27}+\text{18}\right)+\text{105}\left(\text{810}÷\text{18}\right)$

 Key Display Reads Operate inside the parentheses Type 27 27 Press + 27 Type 18 18 Press = 45 Multiply by 42. Press × 45 Type 42 42 Press = 1890

Place this result into memory by pressing the memory key.

 Key Display Reads Now operate in the other parentheses. Type 810 810 Press ÷ 810 Type 18 18 Press = 45 Now multiply by 105. Press × 45 Type 105 105 Press = 4725 We are now ready to add these two quantities together. Press + 4725 Press the memory recall key. 1890 Press = 6615

Thus, $\text{42}\left(\text{27}+\text{18}\right)+\text{105}\left(\text{810}÷\text{18}\right)=6,\text{615}$

${\text{16}}^{4}+{\text{37}}^{3}$

 Nonscientific Calculators Key Display Reads Type 16 16 Press × 16 Type 16 16 Press × 256 Type 16 16 Press × 4096 Type 16 16 Press = 65536 Press the memory key Type 37 37 Press × 37 Type 37 37 Press × 1396 Type 37 37 Press × 50653 Press + 50653 Press memory recall key 65536 Press = 116189
 Calculators with ${y}^{x}$ Key Key Display Reads Type 16 16 Press ${y}^{x}$ 16 Type 4 4 Press = 4096 Press + 4096 Type 37 37 Press ${y}^{x}$ 37 Type 3 3 Press = 116189

Thus, ${\text{16}}^{4}+{\text{37}}^{3}=\text{116},\text{189}$

We can certainly see that the more powerful calculator simplifies computations.

Nonscientific calculators are unable to handle calculations involving very large numbers.

$\text{85612}\cdot \text{21065}$

 Key Display Reads Type 85612 85612 Press × 85612 Type 21065 21065 Press =

This number is too big for the display of some calculators and we'll probably get some kind of error message. On some scientific calculators such large numbers are coped with by placing them in a form called "scientific notation." Others can do the multiplication directly. (1803416780)

## Practice set d

Use a calculator to find each value.

$9,\text{285}+\text{86}\left(\text{49}\right)$

13,499

$\text{55}\left(\text{84}-\text{26}\right)+\text{120}\left(\text{512}-\text{488}\right)$

6,070

${\text{106}}^{3}-{\text{17}}^{4}$

1,107,495

$6,{\text{053}}^{3}$

This number is too big for a nonscientific calculator. A scientific calculator will probably give you $2\text{.}\text{217747109}×{\text{10}}^{\text{11}}$

## Exercises

For the following problems, find each value. Check each result with a calculator.

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

26

$\text{18}+7\cdot \left(4-1\right)$

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

46

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

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

1

$\text{98}÷2÷{7}^{2}$

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

0

$\sqrt{9}+\text{14}$

$\sqrt{\text{100}}+\sqrt{\text{81}}-{4}^{2}$

3

$\sqrt[3]{8}+8-2\cdot 5$

$\sqrt[4]{\text{16}}-1+{5}^{2}$

26

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

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

97

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

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

29

$\text{10}\cdot \left[8+2\cdot \left(6+7\right)\right]$

$\text{21}÷7÷3$

1

${\text{10}}^{2}\cdot 3÷{5}^{2}\cdot 3-2\cdot 3$

$\text{85}÷5\cdot 5-\text{85}$

0

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

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

90

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

$2\cdot \left\{\left(7+7\right)+6\cdot \left[4\cdot \left(8+2\right)\right]\right\}$

508

$0+\text{10}\left(0\right)+\text{15}\cdot \left\{4\cdot 3+1\right\}$

$\text{18}+\frac{7+2}{9}$

19

$\left(4+7\right)\cdot \left(8-3\right)$

$\left(6+8\right)\cdot \left(5+2-4\right)$

144

$\left(\text{21}-3\right)\cdot \left(6-1\right)\cdot \left(7\right)+4\left(6+3\right)$

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

1

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

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

52

${3}^{4}+{2}^{4}\cdot \left(1+5\right)$

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

25,001

$\left(7\right)\cdot \left(\text{16}\right)-{3}^{4}+{2}^{2}\cdot \left({1}^{7}+{3}^{2}\right)$

$\frac{{2}^{3}-7}{{5}^{2}}$

$\frac{1}{\text{25}}$

$\frac{{\left(1+6\right)}^{2}+2}{3\cdot 6+1}$

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

14

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

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

0

$\frac{{6}^{3}-2\cdot {\text{10}}^{2}}{{2}^{2}}+\frac{\text{18}\left({2}^{3}+{7}^{2}\right)}{2\left(\text{19}\right)-{3}^{3}}$

$2\cdot \left\{6+\left[{\text{10}}^{2}-6\sqrt{\text{25}}\right]\right\}$

152

$\text{181}-3\cdot \left(2\sqrt{\text{36}}+3\sqrt[3]{\text{64}}\right)$

$\frac{2\cdot \left(\sqrt{\text{81}}-\sqrt[3]{\text{125}}\right)}{{4}^{2}-\text{10}+{2}^{2}}$

$\frac{4}{5}$

## Exercises for review

( [link] ) The fact that 0 + any whole number = that particular whole number is an example of which property of addition?

( [link] ) Find the product. $4,\text{271}×\text{630}$ .

2,690,730

( [link] ) In the statement $\text{27}÷3=9$ , what name is given to the result 9?

( [link] ) What number is the multiplicative identity?

1

( [link] ) Find the value of ${2}^{4}$ .

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