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

 Page 2 / 2

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

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

## 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

$\sqrt{8}+8-2\cdot 5$

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

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

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

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

$\frac{4}{5}$

## 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 ${2}^{4}$ .

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
hi
Loga
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
7hours 36 min - 4hours 50 min By Richley Crapo By Madison Christian By By Melinda Salzer By Jesenia Wofford By Jessica Collett By Stephen Voron By OpenStax By OpenStax By OpenStax