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Key equations

Rules of Exponents
For nonzero real numbers a and b and integers m and n
Product rule a m a n = a m + n
Quotient rule a m a n = a m n
Power rule ( a m ) n = a m n
Zero exponent rule a 0 = 1
Negative rule a n = 1 a n
Power of a product rule ( a b ) n = a n b n
Power of a quotient rule ( a b ) n = a n b n

Key concepts

  • Products of exponential expressions with the same base can be simplified by adding exponents. See [link] .
  • Quotients of exponential expressions with the same base can be simplified by subtracting exponents. See [link] .
  • Powers of exponential expressions with the same base can be simplified by multiplying exponents. See [link] .
  • An expression with exponent zero is defined as 1. See [link] .
  • An expression with a negative exponent is defined as a reciprocal. See [link] and [link] .
  • The power of a product of factors is the same as the product of the powers of the same factors. See [link] .
  • The power of a quotient of factors is the same as the quotient of the powers of the same factors. See [link] .
  • The rules for exponential expressions can be combined to simplify more complicated expressions. See [link] .
  • Scientific notation uses powers of 10 to simplify very large or very small numbers. See [link] and [link] .
  • Scientific notation may be used to simplify calculations with very large or very small numbers. See [link] and [link] .

Section exercises

Verbal

Is 2 3 the same as 3 2 ? Explain.

No, the two expressions are not the same. An exponent tells how many times you multiply the base. So 2 3 is the same as 2 × 2 × 2 , which is 8. 3 2 is the same as 3 × 3 , which is 9.

When can you add two exponents?

What is the purpose of scientific notation?

It is a method of writing very small and very large numbers.

Explain what a negative exponent does.

Numeric

For the following exercises, simplify the given expression. Write answers with positive exponents.

9 2

81

15 −2

3 2 × 3 3

243

4 4 ÷ 4

( 2 2 ) −2

1 16

( 5 8 ) 0

11 3 ÷ 11 4

1 11

6 5 × 6 −7

( 8 0 ) 2

1

5 −2 ÷ 5 2

For the following exercises, write each expression with a single base. Do not simplify further. Write answers with positive exponents.

4 2 × 4 3 ÷ 4 −4

4 9

6 12 6 9

( 12 3 × 12 ) 10

12 40

10 6 ÷ ( 10 10 ) −2

7 −6 × 7 −3

1 7 9

( 3 3 ÷ 3 4 ) 5

For the following exercises, express the decimal in scientific notation.

0.0000314

3.14 × 10 5

148,000,000

For the following exercises, convert each number in scientific notation to standard notation.

1.6 × 10 10

16,000,000,000

9.8 × 10 −9

Algebraic

For the following exercises, simplify the given expression. Write answers with positive exponents.

a 3 a 2 a

a 4

m n 2 m −2

( b 3 c 4 ) 2

b 6 c 8

( x −3 y 2 ) −5

a b 2 ÷ d −3

a b 2 d 3

( w 0 x 5 ) −1

m 4 n 0

m 4

y −4 ( y 2 ) 2

p −4 q 2 p 2 q −3

q 5 p 6

( l × w ) 2

( y 7 ) 3 ÷ x 14

y 21 x 14

( a 2 3 ) 2

5 2 m ÷ 5 0 m

25

( 16 x ) 2 y −1

2 3 ( 3 a ) −2

72 a 2

( m a 6 ) 2 1 m 3 a 2

( b −3 c ) 3

c 3 b 9

( x 2 y 13 ÷ y 0 ) 2

( 9 z 3 ) −2 y

y 81 z 6

Real-world applications

To reach escape velocity, a rocket must travel at the rate of 2.2 × 10 6 ft/min. Rewrite the rate in standard notation.

A dime is the thinnest coin in U.S. currency. A dime’s thickness measures 1.35 × 10 −3 m. Rewrite the number in standard notation.

0.00135 m

The average distance between Earth and the Sun is 92,960,000 mi. Rewrite the distance using scientific notation.

A terabyte is made of approximately 1,099,500,000,000 bytes. Rewrite in scientific notation.

1.0995 × 10 12

The Gross Domestic Product (GDP) for the United States in the first quarter of 2014 was $ 1.71496 × 10 13 . Rewrite the GDP in standard notation.

One picometer is approximately 3.397 × 10 −11 in. Rewrite this length using standard notation.

0.00000000003397 in.

The value of the services sector of the U.S. economy in the first quarter of 2012 was $10,633.6 billion. Rewrite this amount in scientific notation.

Technology

For the following exercises, use a graphing calculator to simplify. Round the answers to the nearest hundredth.

( 12 3 m 33 4 −3 ) 2

12,230,590,464 m 66

17 3 ÷ 15 2 x 3

Extensions

For the following exercises, simplify the given expression. Write answers with positive exponents.

( 3 2 a 3 ) −2 ( a 4 2 2 ) 2

a 14 1296

( 6 2 −24 ) 2 ÷ ( x y ) −5

m 2 n 3 a 2 c −3 a −7 n −2 m 2 c 4

n a 9 c

( x 6 y 3 x 3 y −3 y −7 x −3 ) 10

( ( a b 2 c ) −3 b −3 ) 2

1 a 6 b 6 c 6

Avogadro’s constant is used to calculate the number of particles in a mole. A mole is a basic unit in chemistry to measure the amount of a substance. The constant is 6.0221413 × 10 23 . Write Avogadro’s constant in standard notation.

Planck’s constant is an important unit of measure in quantum physics. It describes the relationship between energy and frequency. The constant is written as 6.62606957 × 10 −34 . Write Planck’s constant in standard notation.

0.000000000000000000000000000000000662606957

Questions & Answers

Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
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Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:  OpenStax, Selected topics in algebra. OpenStax CNX. Sep 02, 2015 Download for free at http://legacy.cnx.org/content/col11877/1.2
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