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General, Organic, and

Biochemistry for

Nursing and Allied

Health

SCHAUM'S

outlines

This page intentionally left blank

General, Organic, and

Biochemistry for

Nursing and Allied

Health

Second Edition

George Odian, Ph.D.

Professor of Chemistry

The College of Staten Island

City University of New York

Ira Blei, Ph.D.

Professor of Chemistry

The College of Staten Island

City University of New York

Schaum’s Outline Series

New York Chicago San Francisco Lisbon London

Madrid Mexico City Milan New Delhi San Juan

Seoul Singapore Sydney Toronto

SCHAUM'S

outlines

MC

Graw

Hill

Copyright © 2009, l994 by The McGraw-Hill Companies, Inc. All rights reserved. Except as permitted under the United States Copyright Act of 1976, no

part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written

permission of the publisher.

ISBN: 978-0-07-162351-7

MHID: 0-07-162351-5

The material in this eBook also appears in the print version of this title: ISBN: 978-0-07-161165-7, MHID: 0-07-161165-7.

All trademarks are trademarks of their respective owners. Rather than put a trademark symbol after every occurrence of a trademarked name, we use names

in an editorial fashion only, and to the benefit of the trademark owner, with no intention of infringement of the trademark. Where such designations appear

in this book, they have been printed with initial caps.

McGraw-Hill eBooks are available at special quantity discounts to use as premiums and sales promotions, or for use in corporate training programs. To

contact a representative please e-mail us at [email protected].

TERMS OF USE

This is a copyrighted work and The McGraw-Hill Companies, Inc. (“McGraw-Hill”) and its licensors reserve all rights in and to the work. Use of this work

is subject to these terms. Except as permitted under the Copyright Act of 1976 and the right to store and retrieve one copy of the work, you may not decom￾pile, disassemble, reverse engineer, reproduce, modify, create derivative works based upon, transmit, distribute, disseminate, sell, publish or sublicense the

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THE WORK IS PROVIDED “AS IS.” McGRAW-HILL AND ITS LICENSORS MAKE NO GUARANTEES OR WARRANTIES AS TO THE

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v

Preface

This book is intended for students who are preparing for careers in health fields such as nursing, physical

therapy, podiatry, medical technology, agricultural science, public health, and nutrition. The chemistry

courses taken by these students typically include material covered in general chemistry, organic chemistry,

and biochemistry, compressed into a 1-year period. Because of this broad requirement, many students feel

overwhelmed. To help them understand and assimilate so much diverse material, we offer the outline of

these topics presented in the text that follows. Throughout we have kept theoretical discussions to a mini￾mum in favor of presenting key topics as questions to be answered and problems to be solved. The book

can be used to accompany any standard text and to supplement lecture notes. Studying for exams should

be much easier with this book at hand.

The solved problems serve two purposes. First, interspersed with the text, they illustrate, comment on,

and support the fundamental principles and theoretical material introduced. Second, as additional solved

problems and supplementary problems at the end of each chapter, they test a student’s mastery of the mate￾rial and, at the same time, provide step-by-step solutions to the kinds of problems likely to be encountered

on examinations.

No assumptions have been made regarding student knowledge of the physical sciences and mathematics;

such background material is provided where required. SI units are used as consistently as possible. How￾ever, non-SI units that remain in common use, such as liter, atmosphere, and calorie, will be found where

appropriate.

The first chapter emphasizes the current method employed in mathematical calculations, viz., factor￾label analysis. In the section on chemical bonding, although molecular orbitals are discussed, VSEPR

theory (valence-shell electron-pair repulsion theory) is emphasized in characterizing three-dimensional

molecular structure. The discussion of nuclear processes includes material on modern spectroscopic methods

of noninvasive anatomical visualization.

The study of organic chemistry is organized along family lines. To simplify the learning process, the

structural features, physical properties, and chemical behavior of each family are discussed from the view￾point of distinguishing that family from other families with an emphasis on those characteristics that are

important for the consideration of biologically important molecules.

The study of biochemistry includes chapters on the four important families of biochemicals—

carbohydrates, lipids, proteins, and nucleic acids—with an emphasis on the relationship between chemical

structure and biological function for each. These chapters are followed by others on intermediary metabo￾lism and human nutrition. In the discussion of all these topics we have emphasized physiological questions

and applications where possible.

We would like to thank Charles A. Wall (Senior Editor), Anya Kozorez (Sponsoring Editor), Kimberly-Ann

Eaton (Associate Editor), Tama Harris McPhatter (Production Supervisor), and Frank Kotowski, Jr. (Editing

Supervisor) at McGraw-Hill Professional, and Vasundhara Sawhney (Project Manager) at International

Typesetting & Composition for their encouragement and conscientious and professional efforts in bringing

this book to fruition.

The authors welcome comments from readers at [email protected] and [email protected].

GEORGE ODIAN

IRA BLEI

This page intentionally left blank

vii

Contents

CHAPTER 1 Chemistry and Measurement 1

1.1 Introduction. 1.2 Measurement and the metric system. 1.3 Scien￾tific notation (1.3.1 Logarithms). 1.4 Significant figures. 1.5 Significant

figures and calculations. 1.6 Measurement and error. 1.7 Factor-label

method. 1.8 Mass, volume, density, temperature, heat, and other forms

of energy (1.8.1 Mass, 1.8.2 Volume, 1.8.3 Density, 1.8.4 Temperature,

1.8.5 Heat, 1.8.6 Other forms of energy).

CHAPTER 2 Atomic Structure and The Periodic Table 26

2.1 The atomic theory. 2.2 Atomic masses. 2.3 Atomic structure.

2.4 Isotopes. 2.5 The periodic table. 2.6 Atomic structure and periodicity.

CHAPTER 3 Compounds and Chemical Bonding 45

3.1 Introduction. 3.2 Nomenclature (3.2.1 Binary ionic compounds,

3.2.2 Polyatomic ions, 3.2.3 Covalent compounds). 3.3 Ionic bonds.

3.4 Covalent bonds. 3.5 Lewis structures. 3.6 Three-dimensional mo￾lecular structures.

CHAPTER 4 Chemical Calculations 69

4.1 Chemical formulas and formula masses. 4.2 The mole. 4.3 Avogadro’s

number. 4.4 Empirical formulas and percent composition. 4.5 Molecular

formula from empirical formula and molecular mass. 4.6 Balancing chemical

equations. 4.7 Stoichiometry.

CHAPTER 5 Physical Properties of Matter 88

5.1 Introduction. 5.2 Kinetic-molecular theory. 5.3 Cohesive forces.

5.4 The gaseous state (5.4.1 Gas pressure, 5.4.2 The gas laws, 5.4.3 Boyle’s

law, 5.4.4 Charles’ law, 5.4.5 Combined gas laws, 5.4.6 The ideal gas law,

5.4.7 The ideal gas law and molecular mass, 5.4.8 Dalton’s law of partial

pressures). 5.5 Liquids (5.5.1 Liquids and vapor pressure, 5.5.2 Viscosity

of liquids, 5.5.3 Surface tension). 5.6 Solids.

CHAPTER 6 Concentration and its Units 109

6.1 Introduction. 6.2 Percent concentration. 6.3 Molarity. 6.4 Molality.

viii Contents

CHAPTER 7 Solutions 119

7.1 Solutions as mixtures. 7.2 Solubility (7.2.1 Solubility of gases,

7.2.2 Solubility of solids). 7.3 Water. 7.4 Dilution. 7.5 Neutralization

and titration. 7.6 Colligative properties, diffusion, and membranes

(7.6.1 Osmotic pressure, 7.6.2 Freezing point depression and boiling point

elevation).

CHAPTER 8 Chemical Reactions 134

8.1 Introduction. 8.2 Chemical kinetics (8.2.1 Collision theory, 8.2.2 Heat

of reaction and activation energy, 8.2.3 Reaction rates, 8.2.4 Effect of con￾centration on reaction rate, 8.2.5 Effect of temperature on reaction rate,

8.2.6 Effect of catalysts on reaction rate). 8.3 Chemical equilibrium

(8.3.1 Equilibrium constants). 8.4 Le Chatelier principle. 8.5 Oxidation￾reduction reactions (8.5.1 Oxidation states, 8.5.2 Balancing redox reactions,

8.5.3 Combustion reactions).

CHAPTER 9 Aqueous Solutions of Acids, Bases, and Salts 155

9.1 Acid-base theories (9.1.1 Acids and bases according to Arrhenius,

9.1.2 Acids and bases according to Brønsted and Lowry, 9.1.3 Lewis acids

and bases). 9.2 Water reacts with water. 9.3 Acids and bases: strong versus

weak. 9.4 pH, a measure of acidity. 9.5 pH and weak acids and bases.

9.6 Polyprotic acids. 9.7 Salts and hydrolysis. 9.8 Buffers and buffer

solutions. 9.9 Titration. 9.10 Normality.

Organic Chemistry

CHAPTER 10 Nuclear Chemistry and Radioactivity 177

10.1 Radioactivity (10.1.1 Radioactive emissions, 10.1.2 Radioactive decay,

10.1.3 Radioactive series, 10.1.4 Transmutation, 10.1.5 Nuclear fission,

10.1.6 Nuclear fusion, 10.1.7 Nuclear energy). 10.2 Effects of radiation.

10.3 Detection. 10.4 Units. 10.5 Applications.

CHAPTER 11 Organic Compounds; Saturated Hydrocarbons 193

11.1 Organic chemistry. 11.2 Molecular and structural formulas.

11.3 Families of organic compounds, functional groups. 11.4 Alkanes.

11.5 Writing structural formulas. 11.6 Constitutional isomers.

11.7 Nomenclature (11.7.1 Alkyl groups, 11.7.2 IUPAC nomenclature,

11.7.3 Other names). 11.8 Cycloalkanes. 11.9 Physical properties

(11.9.1 Boiling and melting points, 11.9.2 Solubility). 11.10 Chemical reac￾tions (11.10.1 Halogenation, 11.10.2 Combustion).

CHAPTER 12 Unsaturated Hydrocarbons: Alkenes, Alkynes, Aromatics 220

12.1 Alkenes. 12.2 The carbon-carbon double bond. 12.3 Constitutional

isomerism in alkenes. 12.4 Nomenclature of alkenes. 12.5 Cis-trans isomers

(12.5.1 Alkenes, 12.5.2 Cycloalkanes). 12.6 Chemical reactions of alkenes

(12.6.1 Addition, 12.6.2 Mechanism of addition reactions, 12.6.3 Polymerization,

12.6.4 Oxidation). 12.7 Alkynes. 12.8 Aromatics. 12.9 Nomenclature of

aromatic compounds. 12.10 Reactions of benzene.

Contents ix

CHAPTER 13 Alcohols, Phenols, Ethers, and Thioalcohols 248

13.1 Alcohols. 13.2 Constitutional isomerism in alcohols. 13.3 Nomen￾clature of alcohols. 13.4 Physical properties of alcohols. 13.5 Chemical

reactions of alcohols (13.5.1 Acid-base properties, 13.5.2 Dehydration,

13.5.3 Oxidation). 13.6 Phenols. 13.7 Ethers. 13.8 Thioalcohols.

CHAPTER 14 Aldehydes and Ketones 271

14.1 Structure of aldehydes and ketones. 14.2 Constitutional isomer￾ism in aldehydes and ketones. 14.3 Nomenclature of aldehydes and

ketones. 14.4 physical properties of aldehydes and ketones. 14.5 Oxi￾dation of aldehydes and ketones. 14.6 Reduction of aldehydes and ketones.

14.7 Reaction of aldehydes and ketones with alcohol.

CHAPTER 15 Carboxylic Acids, Esters, and Related Compounds 288

15.1 Structure of carboxylic acids. 15.2 Nomenclature of carboxylic acids.

15.3 Physical properties of carboxylic acids. 15.4 Acidity of carboxylic

acids. 15.5 Soaps and detergents. 15.6 Conversion of carboxylic acids to

esters. 15.7 Nomenclature and physical properties of esters. 15.8 Chemical

reactions of esters. 15.9 Carboxylic acid anhydrides, halides and amides.

15.10 Phosphoric acid anhydrides and esters.

CHAPTER 16 Amines and Amides 305

16.1 Amines. 16.2 Constitutional isomerism in amines. 16.3 Nomen￾clature of amines. 16.4 Physical properties of amines. 16.5 Chemical

reactions of amines (16.5.1 Basicity, 16.5.2 Nucleophilic substitution on alkyl

halides). 16.6 Conversion of amines to amides. 16.7 Nomenclature and

physical properties of amides. 16.8 Chemical reactions of amides.

CHAPTER 17 Stereoisomerism 322

17.1 Review of isomerism (17.l.l Constitutional isomers, 17.1.2 Geo￾metrical isomers). 17.2 Enantiomers. 17.3 Nomenclature and prop￾erties of enantiomers (17.3.1 Nomenclature, 17.3.2 Physical properties,

17.3.3 Chemical properties). 17.4 Compounds with more than one stereo￾center.

Biochemistry

CHAPTER 18 Carbohydrates 339

18.1 Monosaccharides. 18.2 Cyclic hemiacetal and hemiketal structures.

18.3 Properties and reactions of monosaccharides. 18.4 Disaccharides.

18.5 Polysaccharides.

CHAPTER 19 Lipids 361

19.1 Introduction. 19.2 Fatty acids. 19.3 Triacylglycerols (19.3.1 Struc￾ture and physical properties, 19.3.2 Chemical reactions). 19.4 Waxes.

19.5 Phospholipids. 19.6 Sphingolipids. 19.7 Nonhydrolyzable lipids.

19.8 Cell membranes. 19.9 Lipids and health.

x Contents

CHAPTER 20 Proteins 383

20.1 Amino acids. 20.2 Peptide formation. 20.3 Protein structure and

function (20.3.1 Protein shape, 20.3.2 Fibrous proteins, 20.3.3 Globular pro￾teins, 20.3.4 Denaturation).

CHAPTER 21 Nucleic Acids and Heredity 13

21.1 Nucleotides. 21.2 Nucleic acids (21.2.1 Formation of nucleic acids;

21.2.2 Secondary, tertiary, and quaternary structures of DNA; 21.2.3 Secondary,

tertiary, and quaternary structures of RNA). 21.3 Flow of genetic informa￾tion (21.3.1 Replication, 21.3.2 Transcription, 21.3.3 Translation). 21.4 Other

aspects of nucleic acids and protein synthesis (21.4.1 Mutations, 21.4.2 Anti￾bio tics, 21.4.3 Viruses, 21.4.4 Recombinant DNA technology).

CHAPTER 22 Metabolic Systems 441

22.1 Introduction. 22.2 Enzymes, cofactors, and coenzymes. 22.3 Meta￾bolism of carbohydrates. 22.4 Metabolism of lipids. 22.5 Metabolism of

amino acids. 22.6 Energy yield from catabolism.

CHAPTER 23 Digestion, Nutrition, and Gas Transport 467

23.1 Digestion. 23.2 Nutrition (23.2.1 Carbohydrates, 23.2.2 Proteins,

23.2.3 Fats, 23.2.4 Vitamins, 23.2.5 Minerals). 23.3 Metabolic gas transport

(23.3.1 Oxygen transport, 23.3.2 Carbon dioxide transport).

APPENDIX A Basic and Derived SI Units and Conversion Factors 482

APPENDIX B Table of Atomic Masses 483

APPENDIX C Periodic Table 484

INDEX 485

1

Chemistry and Measurement

CHAPTER 1

1.1 INTRODUCTION

Chemistry is the study of matter and energy and the interactions between them. This is an

extremely broad and inclusive definition, but quite an accurate one. There is no aspect of the

description of the material universe which does not depend on chemical concepts, both practical and

theoretical.

Although chemistry is as old as the history of humankind, it remained a speculative and somewhat

mysterious art until about 300 years ago. At that time it became clear that matter comes in many

different forms and kinds; therefore some kind of classification was needed, if only to organize data.

There was red matter and white matter, liquid matter and solid matter, but it did not take long to

realize that such broad qualitative descriptions, although important, were not sufficient to differentiate

one kind of matter from another. Additional criteria, now called properties, were required. It was

found that these properties could be separated into two basic classes: physical and chemical. Changes

in physical properties involve only changes in form or appearance of a substance; its fundamental

nature remains the same. For example, the freezing of water involves only its conversion from liquid

to solid. The fact that its fundamental nature remains the same is easily demonstrated by melting the

ice. By passing an electric current through water, however, two new substances are created: hydrogen

and oxygen. The fundamental nature of water is changed—it is no longer water, but has been

transformed into new substances through chemical change.

Without knowing anything about the fundamental nature of matter, chemists were also able to

establish that matter could be separated into simpler and simpler substances through physical

separation methods (e.g., distillation, solubility) and through chemical reactivity. They developed

methods for measuring physical properties such as density, hardness, color, physical state, and melting

and boiling points to help them decide when these operations could no longer change the nature of

the substance. From these considerations, another classification scheme emerged, based on composi￾tion. In this scheme, matter is divided into two general classes: pure substances and matures.

There are two kinds of pure substances: elements and compounds. An element is a substance that

cannot be separated into simpler substances by ordinary chemical methods. Nor can it be created by

combining simpler substances. All the matter in the universe is composed of one or more of these

fundamental substances. When elements are combined, they form compounds—substances having

definite, fixed proportions of the combined elements with none of the properties of the individual

elements, but with their own unique set of new physical and chemical properties.

In contrast to the unique properties of compounds, the properties of mixtures are variable and

depend on composition. An example is sugar in water. The most recognizable property of this mixture

is its sweetness, which varies depending on its composition (the amount of sugar dissolved in the

water). A mixture is then composed of at least two pure substances. In addition, there are two kinds of

mixtures. Homogeneous mixtures, or solutions, are visually uniform (microscopically as well) through￾out the sample. Heterogeneous mixtures reveal visual differences throughout the sample (pepper and

salt, sand and water, whole blood).

1.2 MEASUREMENT AND THE METRIC SYSTEM

Most of the above considerations depended upon the establishment of the quantitative properties

of matter. This required a system of units, and devices for measurement. The measuring device most

familiar to you is probably the foot ruler or yardstick, now being replaced by the centimeter ruler and

the meter stick, both of which measure length. Other devices measure mass, temperature, volume, etc.

The units for these measures have been established by convention and promulgated by authority. This

2 CHAPTER 1 Chemistry and Measurement

assures that a meter measured anywhere in the world is the same as any other meter. This

standardization of units for measurement is fundamental to the existence of modern technological

society. Imagine the consequences if a cubic centimeter of insulin solution in Albuquerque were not

the same as a cubic centimeter of insulin solution in Wichita!

The standardized measurement units used in science and technology today are known as the

metric system. It was originally established in 1790 by the French National Academy, and has

undergone changes since then. The fundamental or base units of the modern metric system (SI for

Systeme International d'Unites) are found in Table 1-1. In chemistry, for the most part, you will

encounter the first five of these. All other units are derived from these fundamental units. For

example:

square meters (m2

) = area

cubic meters (m3

) = volume

density = kilograms/cubic meters (kg/m3

)

velocity = meters/second (m/s)

Older, non-Si units are in common use, and some of these are shown in Table 1-2.

Table 1-1. Fundamental Units of the Modern Metric System

Fundamental Quantity

Length

Mass

Temperature

Time

Amount of substance

Electric current

Luminous intensity

Unit Name

meter

kilogram

kelvin

second

mole

ampere

candela

Symbol

m

kg

K

s

mol

A

cd

Table 1-2. Non-Si Units in Common Use

Quantity

Length

Volume

Energy

Unit

Angstrom

Liter

Calorie

Symbol

A

L

cal

SI Definition

10-10m

10-3

m3

kg-m2

/s2

*

SI Name

0.1 nanometers (nm)

1 decimeter3

(dm3

)

4.184 joules (J)

*A dot (center point) will be used in this book to denote multiplication in derived units.

All of these units can be expressed in parts of or multiples of 10. The names of these multiples are

created by the use of prefixes of Greek and Latin origin. This is best illustrated by Table 1-3. These

symbols can be used with any kind of unit to denote size, e.g., nanosecond (ns), millimol (mmol),

kilometer (km). Some of the properties commonly measured in the laboratory will be discussed in

detail in a following section.

Problem 1.1. Express (a) 0.001 second (s); (b) 0.99 meter (m); (c) 0.186 liter (L) in more convenient units.

Ans. (a) 1 millisecond (ms); (b) 99 centimeters (cm); (c) 186 milliliters (mL).

CHAPTER 1 Chemistry and Measurement 3

Table 1-3. Names Used to Express Metric Units in Multiples of 10

Multiple or Part of 10

1,000,000

1000

100

10

0.1

0.01

0.001

0.000001

0.000000001

Prefix

mega

kilo

hecto

deka

deci

centi

milli

micro

nano

Symbol

M

k

h

da

d

c

m

M

n

13 SCIENTIFIC NOTATION

It is inconvenient to be limited to decimal representations of numbers. In chemistry, very large

and very small numbers are commonly used. The number of atoms in about 12 grams (g) of carbon is

represented by 6 followed by 23 zeros. Atoms typically have dimensions of parts of nanometers, i.e.,

10 decimal places. A far more practical method of representation is called scientific or exponential

notation. A number expressed in scientific notation is a number between 1 and 10 which is then

multiplied by 10 raised to a whole number power. The number between 1 and 10 is called the

coefficient, and the factor of 10 raised to a whole number is called the exponential factor.

Problem 1.2. Express the numbers 1, 10, 100, and 1000 in scientific notation.

Ans. We must first choose a number between 1 and 10 for each case. In this example, the number is the

same for all, 1. This number must be multiplied by 10 raised to a power which is a whole number.

There are two rules to remember; (a) any number raised to the zero power is equal to one (1), and

(fr) when numbers are multiplied, the exponents must be added. Examples are:

1 = 1 X 10°

10=1 xlO1

100=lxl01

xlO' = lxl02

1000 = 1 x 10' x 10' x 10' = 1 x 103

Note that in each case, the whole number to which 10 is raised is equal to the number of places the

decimal point was moved to the left.

Problem 13. Express the number 4578 in scientific notation.

Ans. In this case, the number between 1 and 10 must be 4.578, which is also the result of moving the

decimal point three places to the left. In scientific notation, 4578 is written 4.578 X 103

.

Problem 1.4. Express the numbers 0.1, 0.01, 0.001, and 0.0001 in scientific notation.

Ans. The process leading to scientific notation for decimals involves expressing these numbers as

fractions, then recalling the algebraic rule that the reciprocal of any quantity X (which includes

units), that is I/A', may be expressed as X~l

. For example, l/5 = 5~', and l/cm = cm"1

.

4 CHAPTER 1 Chemistry and Measurement

Note that in each case, the whole number to which 10 is raised is equal to the number of places the

decimal point is moved to the right.

Therefore, moving the decimal point to the right requires a minus sign before the power of 10,

and moving the decimal point to the left requires a plus sign before the power of 10.

Problem 1.5. Express the number 0.00352 in scientific notation.

Ans. To obtain a number between 1 and 10, we move the decimal point three places to the right. This

yields 3.52, which is then multiplied by 10 raised to (— 3) since we moved the decimal to the right:

3.52 x 10~3

. (It is useful to remember that any number smaller than 1.0 must be raised to a

negative power of 10, and conversely any number greater than 1.0 must be raised to a positive

power of 10.)

Problem 1.6. How many ways can the number 0.00352 be represented in scientific notation?

Ans. Since the value of the number must remain constant, the product of the coefficient and the

exponential factor must remain constant, but the size of the coefficient can be varied as long as the

value of the exponential factor is also properly modified. The modification is accomplished by

either multiplying the coefficient by 10 and dividing the exponent by 10, or dividing the coefficient

by 10 and multiplying the exponential factor by 10. Either process leaves the value of the number

unchanged.

0.00352 = 0.00352 x 10° = 0.0352 X 10~' = 0.352 x KT2

= 3.52 X 10~3

Problem 1.7. Add 2.0 x 103

and 3.4 x 104

.

Ans. When adding (or subtracting) exponential numbers, first convert all exponents to the same value,

then add (or subtract) the coefficients and multiply by the now common exponential factor. If we

write the numbers in nonexponential form, it is easy to see why the rule works:

3.4 x 104

= 34 x 103

= 34,000

2.0 X 103

= 2000

34,000 + 2000 = 36,000

(34 X 103) + (2.0 x 103) = (34 + 2.0) x 103 = 36 X 103 = 3.6 X 104

Problem 1.8. Multiply the numbers 2.02 X 103

and 3.20 X 1Q-2

.

Ans. When exponential numbers are multiplied, the coefficients are multiplied and the exponents are

added. A simplified calculation will show the derivation of this rule:

102

x 103

= 101

X 10' X 101

x 101

X 101

= 10s

Therefore

(2.02 X 103

) X (3.20 X 10-2

) = 6.46 X 101

= 64.6

Therefore, each of the above cases may be written

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