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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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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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