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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

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
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?
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
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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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