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a 0 = 1 size 12{a rSup { size 8{0} } `=`1} {} , ( a 0 ) size 12{` \( a<>0 \) } {}

For example, x 0 = 1 and ( 1,000,000 ) 0 = 1 size 12{x rSup { size 8{0} } `=``1" and " \( "1,000,000" \) rSup { size 8{0} } `=``1} {} .

Note that the base must be a non-zero value. 0 0 is called an indeterminate number, and has no value. This is because 0 0 = 0/0. If one considers 0 = 0 × n (where n can be any number) then it follows that 0/0 = n , where n can be any number – meaning the value of 0/0 cannot be determined.

Examples: application using exponential law 1

  1. 16 0 = 1 size 12{"16" rSup { size 8{0} } =``1} {}
  2. 16 a 0 = 16 size 12{"16"a rSup { size 8{0} } =``"16"} {}
  3. ( 16 + a ) 0 = 1 size 12{ \( "16"+a \) rSup { size 8{0} } =``1} {}
  4. ( 16 ) 0 = 1 size 12{ \( - "16" \) rSup { size 8{0} } =``1} {}
  5. 16 0 = 1 size 12{ - "16" rSup { size 8{0} } =`` - 1} {}

Exponential law 2

Our definition of exponential notation shows that:

a m × a n = a m + n size 12{a rSup { size 8{m} } ` times `a rSup { size 8{n} } `=`a rSup { size 8{m+n} } } {}

That is:

a m a n = 1 a a size 12{a rSup { size 8{m} } cdot a rSup { size 8{n} } `=``1` cdot `a` cdot ` dotslow ` cdot `a } {}  ( m times) 1 a a size 12{` cdot `1` cdot `a` cdot ` dotslow ` cdot ` ital "a "} {}   ( n times)

             = 1 a a size 12{ {}= `1` cdot `a` cdot ` dotslow ` cdot `a" "``} {}     ( m + n times)

             = a m + n size 12{ {}= ital " a" rSup { size 8{m+n} } } {}

For example:

2 7 2 3 = ( 2 2 2 2 2 2 2 ) ( 2 2 2 ) = 2 10 = 2 7 + 3 alignl { stack { size 12{`2 rSup { size 8{7} } cdot 2 rSup { size 8{3} } = \( 2 cdot 2 cdot 2 cdot 2 cdot 2 cdot 2 cdot 2 \) ital " " \( 2 cdot 2 cdot 2 \) } {} #`= 2 rSup { size 8{"10"} } {} # `= 2 rSup { size 8{7+3} } {}} } {}

This simple law illustrates the reason exponentials were originally invented. In the days before calculators, all multiplication had to be done by hand with a pencil and a pad of paper. Multiplication takes a very long time to do and is very tedious. Adding numbers, however, is easy and quick. This law says that adding the exponents of two exponential numbers (of the same base) is the same as multiplying the two numbers together. This means that, for certain numbers, there is no need to actually multiply the numbers together in order to find their multiple. This saved mathematicians a lot of time.

Examples: application using exponential law 2

  1. x 2 x 5 = x 7 size 12{x rSup { size 8{2} } cdot x rSup { size 8{5} } = ital " x" rSup { size 8{7} } } {}
  2. 2x 3 y 5x 2 y 7 = 10 x 5 y 8 size 12{2x rSup { size 8{3} } y cdot 5x rSup { size 8{2} } y rSup { size 8{7} } = "10"x rSup { size 8{5} } y rSup { size 8{8} } } {}
  3. 2 3 2 4 = 2 7 size 12{2 rSup { size 8{3} } cdot 2 rSup { size 8{4} } = 2 rSup { size 8{7} } } {}    (Note that the base (2) stays the same.)
  4. 3 3 2a 3 2 = 3 2a + 3 size 12{3 cdot 3 rSup { size 8{2a} } cdot 3 rSup { size 8{2} } =3 rSup { size 8{2a+3} } } {}

Exponential law 3

a m ÷ a n = a m n size 12{a rSup { size 8{m} } `` div ``a rSup { size 8{n} } `=`a rSup { size 8{m - n} } } {}

We know from Law 2 that a m + n size 12{a rSup { size 8{m+n} } } {} is base a multiplied by itself m times plus a multiplied by itself n times. Law 3 extends this to the case where an exponent is negative.

a m a n = a a a a a a a a size 12{ { {a rSup { size 8{m} } } over {a rSup { size 8{n} } } } `=` { {`a cdot a cdot a` dotsaxis ` cdot a`} over {a cdot a cdot a` dotsaxis ` cdot a} } } {} ( m times ) ( n times ) size 12{ { {` \( m`"times" \) `} over { \( n`"times" \) } } } {}

By factoring out a n size 12{a rSup { size 8{n} } } {} from both numerator and denominator, we are left with

     = a a a a a a a a size 12{``=` { {`a cdot a cdot a dotsaxis cdot a`} over {`a cdot a cdot a dotsaxis cdot a`} } } {} ( m times ) ( n times ) size 12{ { {` \( m`"times" \) `} over { \( n`"times" \) } } } {} a a a a a a a a size 12{ { { - `a cdot a cdot a` dotsaxis cdot a} over { - `a cdot a cdot a` dotsaxis cdot a} } } {} ( n times ) ( n times ) size 12{ { {` \( n`"times" \) `} over { \( n`"times" \) } } } {}

     = a a a a size 12{``=`a cdot a cdot a dotsaxis cdot a`} {}    ( m n times)

     = a m n size 12{``=`a rSup { size 8{m - n} } } {}

For example,

2 7 ÷ 2 3 = 2 2 2 2 2 2 2 2 2 2 = 2 2 2 2 = 2 4 = 2 7 3 alignl { stack { size 12{`2 rSup { size 8{7} } div 2 rSup { size 8{3} } `=` { {2 cdot 2 cdot 2 cdot 2 cdot 2 cdot 2 cdot 2} over {2 cdot 2 cdot 2} } } {} #```````````=``2 cdot 2 cdot 2 cdot 2 {} # ```````````=``2 rSup { size 8{4} } {} #```````````=``2 rSup { size 8{7 - 3} } {} } } {}

Examples: exponential law 3

  1. a 6 a 2 = a 6 2 = a 4 size 12{ { {a rSup { size 8{6} } } over {a rSup { size 8{2} } } } `=`a rSup { size 8{6 - 2} } `=`a rSup { size 8{4} } } {}
  2. 3 2 3 6 = 3 2 6 = 3 4 = 1 3 4 size 12{ { {3 rSup { size 8{2} } } over {3 rSup { size 8{6} } } } ``=``3 rSup { size 8{2 - 6} } ``=``3 rSup { size 8{ - 4} } `=` { {1} over {3 rSup { size 8{4} } } } ```} {}    (Always give the final answer with a positive index)
  3. 32 a 2 4a 8 = 8a 6 = 8 a 6 size 12{ { {"32"a rSup { size 8{2} } } over {4a rSup { size 8{8} } } } `=`8a rSup { size 8{ - 6} } `=` { {8} over {a rSup { size 8{6} } } } } {}
  4. a 3x a 4 = a 3x 4 size 12{ { {a rSup { size 8{3x} } } over {a rSup { size 8{4} } } } `=`a rSup { size 8{3x - 4} } } {}

Exponential law 4

a n = 1 a n , a 0 size 12{a rSup { size 8{ - n} } `= { {1} over {a rSup { size 8{n} } } } ,~`a<>0} {}

Our definition of exponential notation for a negative exponent shows that

a n = 1 ÷ a ÷ ÷ a size 12{a rSup { size 8{ - n} } `=`1` div `a` div ` dotsaxis ` div `a} {}    ( n times)

       = 1 1 a a size 12{ {}=` { {1} over {1` cdot `a` cdot ` dotsaxis ` cdot `a} } } {} ( n times ) size 12{ { {``} over { \( n`"times" \) } } } {}  

       = 1 a n size 12{ {}=` { {1} over {a rSup { size 8{n} } } } } {}

The minus sign in the exponent is just another way of writing that the whole exponential number is to be divided instead of multiplied.

For example, starting with Law 3, take the case of a m n size 12{a rSup { size 8{m - n} } } {} , but where  n>m :

2 2 9 = 2 2 2 9 = 2 2 2 2 2 2 2 2 2 2 2 = 1 2 2 2 2 2 2 2 = 1 2 7 = 2 7 alignl { stack { size 12{`2 rSup { size 8{2 - 9} } `=` { {2 rSup { size 8{2} } } over {2 rSup { size 8{9} } } } `} {} #```````=` { {2` cdot `2} over {2` cdot `2` cdot `2` cdot `2` cdot `2` cdot `2` cdot `2` cdot `2` cdot `2} } {} # ```````= { {1} over {2 cdot 2 cdot 2 cdot 2 cdot 2 cdot 2 cdot 2} } {} #```````= { {1} over {2 rSup { size 8{7} } } } {} # ```````=`2 rSup { size 8{ - 7} } {}} } {}

Examples: exponential law 4

  1. 2 2 = 1 2 2 = 1 4 size 12{2 rSup { size 8{ - 2} } = { {1} over {2 rSup { size 8{2} } } } = { {1} over {4} } } {}
  2. 2 2 3 2 = 1 2 2 3 2 = 1 36 size 12{ { {2 rSup { size 8{ - 2} } } over {3 rSup { size 8{2} } } } = { {1} over {2 rSup { size 8{2} } cdot 3 rSup { size 8{2} } } } = { {1} over {"36"} } } {}
  3. 2 3 3 = 3 2 3 = 27 8 size 12{ left ( { {2} over {3} } right ) rSup { size 8{ - 3} } = left ( { {3} over {2} } right ) rSup { size 8{3} } = { {"27"} over {8} } } {}
  4. m n 4 = mn 4 size 12{ { {m} over {n rSup { size 8{ - 4} } } } = ital "mn" rSup { size 8{4} } } {}
  5. a 3 x 4 a 5 x 2 = x 4 x 2 a 3 a 5 = x 6 a 8 size 12{ { {a rSup { size 8{ - 3} } cdot x rSup { size 8{4} } } over {a rSup { size 8{5} } cdot x rSup { size 8{ - 2} } } } = { {x rSup { size 8{4} } cdot x rSup { size 8{2} } } over {a rSup { size 8{3} } cdot a rSup { size 8{5} } } } = { {x rSup { size 8{6} } } over {a rSup { size 8{8} } } } } {}

Exponential law 5

( ab ) n = a n b n size 12{ \( ital "ab" \) rSup { size 8{n} } `=`a rSup { size 8{n} } b rSup { size 8{n} } } {}

The order in which two real numbers are multiplied together does not matter.

Therefore,

( ab ) n = a b a b a b a b size 12{ \( ital "ab" \) rSup { size 8{n} } `=``a cdot b cdot a cdot b cdot a cdot b cdot `` dotsaxis ` cdot `a cdot b} {}     ( n times)

         = a a a size 12{`=``a` cdot `a` cdot ` dotslow ` cdot `a} {}  ( n times) b b b size 12{` cdot `b` cdot `b` cdot ` dotslow ` cdot `b} {}  ( n times)

          = a n b n size 12{ {}=``a rSup { size 8{n} } b rSup { size 8{n} } } {}

For example:

2 3 4 = ( 2 3 ) ( 2 3 ) ( 2 3 ) ( 2 3 ) = ( 2 2 2 2 ) ( 3 3 3 3 ) = 2 4 3 4 = 2 4 3 4 alignl { stack { size 12{`2` cdot 3 rSup { size 8{4} } = \( 2 cdot 3 \) cdot \( 2 cdot 3 \) cdot \( 2 cdot 3 \) cdot \( 2 cdot 3 \) } {} #`=`` \( 2 cdot 2 cdot 2 cdot 2 \) ` cdot ` \( 3 cdot 3 cdot 3 cdot 3 \) {} # `= 2 rSup { size 8{4} } ` cdot `3 rSup { size 8{4} } {} #`= 2 rSup { size 8{4} } 3 rSup { size 8{4} } {} } } {}

Examples: exponential law 5

  1. ( 2x 2 y ) 3 = 2 3 x 2 × 3 y 3 = 8x 6 y 3 size 12{ \( 2x rSup { size 8{2} } y \) rSup { size 8{3} } `=`2 rSup { size 8{3} } x rSup { size 8{2 times 3} } y rSup { size 8{3} } `=`8x rSup { size 8{6} } y rSup { size 8{3} } } {}
  2. 7a b 3 2 = 49 a 2 b 6 size 12{ left ( { {7a} over {b rSup { size 8{3} } } } right )` rSup { size 8{2} } `=`` { {"49"a rSup { size 8{2} } } over {b rSup { size 8{6} } } } `} {}
  3. ( 5a n 4 ) 3 = 125 a 3n 12 size 12{ \( 5a rSup { size 8{n - 4} } \) rSup { size 8{3} } `=`"125"a rSup { size 8{3n - "12"} } } {}

Exponential law 6

( a m ) n = a mn size 12{ \( a rSup { size 8{m} } \) rSup { size 8{n} } =a rSup { size 8{ ital "mn"} } } {}

We can find the exponential of an exponential just as well as we can for a number, because an exponential is a real number.

( a m ) n = a m a m a m a m size 12{ \( a rSup { size 8{m} } \) rSup { size 8{n} } `=``a rSup { size 8{m} } ` cdot `a rSup { size 8{m} } ` cdot a rSup { size 8{m} } ` cdot `` dotslow ` cdot `a rSup { size 8{m} } } {}     ( n times)

         = a a a size 12{`=``a cdot a cdot dotslow cdot ital "a " } {}       ( m × n times)

          = a mn size 12{ {}= ital " a" rSup { size 8{ ital "mn"} } } {}

For example:

( 2 2 ) 3 = ( 2 2 ) ( 2 2 ) ( 2 2 ) = ( 2 2 ) ( 2 2 ) ( 2 2 ) = 2 6 = 2 2 × 3 alignl { stack { size 12{`` \( 2 rSup { size 8{2} } \) rSup { size 8{3} } = \( 2 rSup { size 8{2} } \) cdot \( 2 rSup { size 8{2} } \) cdot \( 2 rSup { size 8{2} } \) } {} #``````````=`` \( 2 cdot 2 \) ` cdot ` \( 2 cdot 2 \) ` cdot ` \( 2 cdot 2 \) {} # ``````````= 2 rSup { size 8{6} } {} #``````````= 2 rSup { size 8{2 times 3} } {} } } {}

Examples: exponential law 6

  1. ( x 3 ) 4 = x 12 size 12{ \( x rSup { size 8{3} } \) rSup { size 8{4} } `=`x rSup { size 8{"12"} } } {}
  2. [ ( a 4 ) 3 ] 2 = a 24 size 12{ \[ \( a rSup { size 8{4} } \) rSup { size 8{3} } \] rSup { size 8{2} } `=``a rSup { size 8{"24"} } } {}
  3. ( 3 n + 3 ) 2 = 3 2n + 6 size 12{ \( 3 rSup { size 8{n+3} } \) rSup { size 8{2} } `=`3 rSup { size 8{2n+6} } } {}

Module review exercises

Write the following examples using exponential notation.

4 4 size 12{4` cdot `4} {}

4 2

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12 12 size 12{"12"` cdot `"12"} {}

12 2

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9 9 9 9 size 12{9` cdot `9` cdot `9` cdot `9} {}

9 4

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10 10 10 10 10 10 size 12{"10"` cdot `"10"` cdot `"10"` cdot `"10"` cdot `"10"` cdot `"10"} {}

10 6

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826 826 826 size 12{"826"` cdot `"826"` cdot `"826"} {}

826 3

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3021 3021 3021 3021 size 12{"3021"` cdot `"3021"` cdot `"3021" cdot `"3021"} {}

3021 4

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6 6 6 6 size 12{6` cdot `6` cdot `6` cdot dotsaxis ` cdot `6} {}     (85 factors of 6).

6 85

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2 2 2 2 size 12{`2` cdot `2` cdot `2` cdot ` dotsaxis ` cdot `2} {}     (112 factors of 2).

2 112

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For the next examples, expand the terms. (Do not find the actual values).

117 5

117 · 117 · 117 · 117 · 117

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Determine the value of each of the powers.

Simplify as far as possible.

(2x) 3

2 3 · x 3 = 8x 3

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(-2x) 3

(-2) 3 · x 3 = -8x 3

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x 8 x 3 size 12{ { {x rSup { size 8{8} } } over {x rSup { size 8{3} } } } } {}

x 8 3 = x 5 size 12{`x rSup { size 8{8 - 3} } `=`x rSup { size 8{5} } } {}

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25 x 2 5x 8 size 12{` { {"25"x rSup { size 8{2} } } over {5x rSup { size 8{8} } } } } {}

25 5 x 2 8 = 5x 6 = 5 x 6 size 12{` { {"25"} over {5} } x rSup { size 8{2 - 8} } `=`5x rSup { size 8{ - 6} } `=` { {5} over {x rSup { size 8{6} } } } } {}

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(3 -1 +2 -1 ) -1

1 3 1 + 2 1 = 1 1 3 + 1 2 = 1 3 1 + 2 1 = 3 + 2 = 5 size 12{` { {1} over {3 rSup { size 8{ - 1} } `+`2 rSup { size 8{ - 1} } } } `=` { {1} over { { {1} over {3} } `+` { {1} over {2} } } } `=`1` cdot ` left ( { {3} over {1} } `+` { {2} over {1} } right )`=`3`+`2`=`5} {}

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Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
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?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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
Do you know which machine is used to that process?
s.
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
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Source:  OpenStax, Basic math textbook for the community college. OpenStax CNX. Jul 04, 2009 Download for free at http://cnx.org/content/col10726/1.1
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