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x 5 x 2 = x x x x x x x = ( x x ) x x x ( x x ) = x x x = x 3 . Notice that 5 2 = 3 .

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a 8 a 3 = a a a a a a a a a a a = ( a a a ) a a a a a ( a a a ) = a a a a a = a 5 . Notice that 8 3 = 5 .

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Quotient rule for exponents

If x is a real number and n and m are natural numbers,

x n x m = x n m , x 0 .

To divide two exponential quantities having the same nonzero base, subtract the exponent of the denominator from the exponent of the numerator. Keep in mind that the exponential quantities being divided must have the same base for this rule to apply.

Sample set c

Find the following quotients. All exponents are natural numbers.

x 5 x 2 = x 5-2 = x 3 The part in the box is usally done mentally .

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27 a 3 b 6 c 2 3 a 2 b c = 9 a 3 2 b 6 1 c 2 1 = 9 a b 5 c

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15 x 3 x = 5 x

The bases are the same, so we subtract the exponents. Although we don’t know exactly what is, the notation indicates the subtraction.

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Practice set c

Find each quotient

( x + 6 ) 5 ( x + 6 ) 3

( x + 6 ) 2

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26 x 4 y 6 z 2 13 x 2 y 2 z

2 x 2 y 4 z

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When we make the subtraction, n m , in the division x n x m , there are three possibilities for the values of the exponents:

  1. The exponent of the numerator is greater than the exponent of the denominator, that is, n > m . Thus, the exponent, n m , is a natural number.
  2. The exponents are the same, that is, n = m . Thus, the exponent, n m , is zero, a whole number.
  3. The exponent of the denominator is greater than the exponent of the numerator, that is, n < m . Thus, the exponent, n m , is an integer.

Zero as an exponent

In Sample Set C, the exponents of the numerators were greater than the exponents of the denominators. Let’s study the case when the exponents are the same.

When the exponents are the same, say n , the subtraction n n produces 0.

Thus, by the second rule of exponents, x n x n = x n n = x 0 .

But what real number, if any, does x 0 represent? Let’s think for a moment about our experience with division in arithmetic. We know that any nonzero number divided by itself is one.

8 8 = 1 , 43 43 = 1 , 258 258 = 1

Since the letter x represents some nonzero real number, so does x n . Thus, x n x n

represents some nonzero real number divided by itself. Then x n x n = 1 .

But we have also established that if x 0 , x n x n = x 0 . We now have that x n x n = x 0

and x n x n = 1 . This implies that x 0 = 1 , x 0 .

Exponents can now be natural numbers and zero. We have enlarged our collection of numbers that can be used as exponents from the collection of natural numbers to the collection of whole numbers.

Zero as an exponent

If x 0 , x 0 = 1

Any number, other than 0, raised to the power of 0, is 1. 0 0 has no meaning (it does not represent a number).

Sample set d

Find each value. Assume the base is not zero.

2 x 2 x 2 = 2 x 0 = 2 1 = 2

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5 ( x + 4 ) 8 ( x 1 ) 5 5 ( x + 4 ) 3 ( x 1 ) 5 = ( x + 4 ) 8 3 ( x 1 ) 5 5 = ( x + 4 ) 5 ( x 1 ) 0 = ( x + 4 ) 5

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Practice set d

Find each value. Assume the base is not zero.

6 x 4 2 x 3

3 x 4 3 = 3 x

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14 a 7 7 a 2

2 a 7 2 = 2 a 5

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26 x 2 y 5 4 x y 2

13 2 x y 3

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36 a 4 b 3 c 8 8 a b 3 c 6

9 2 a 3 c 2

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51 ( a 4 ) 3 17 ( a 4 )

3 ( a 4 ) 2

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52 a 7 b 3 ( a + b ) 8 26 a 2 b ( a + b ) 8

2 a 5 b 2

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14 x r y p z q 2 x r y h z 5

7 y p h z q 5

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We will study the case where the exponent of the denominator is greater than the exponent of the numerator in Section [link] .

Exercises

Use the product rule and quotient rule of exponents to simplify the following problems. Assume that all bases are nonzero and that all exponents are whole numbers.

12 x y 3 z 2 4 x 2 y 2 z 3 x

144 x 4 y 5 z 3

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( 4 x 2 ) ( 8 x y 3 )

32 x 3 y 3

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( 2 x y ) ( 3 y ) ( 4 x 2 y 5 )

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( 1 4 a 2 b 4 ) ( 1 2 b 4 )

1 8 a 2 b 8

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( 3 8 ) ( 16 21 x 2 y 3 ) ( x 3 y 2 )

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15 x 20 y 24 z 4 5 x 19 y z

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24 x 4 y 4 z 0 w 8 9 x y w 7

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a 4 b 6 ( a 10 b 16 a 5 b 7 )

a 9 b 15

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3 a 2 b 3 ( 14 a 2 b 5 2 b )

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( x + 3 y ) 11 ( 2 x 1 ) 4 ( x + 3 y ) 3 ( 2 x 1 )

( x + 3 y ) 8 ( 2 x 1 ) 3

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40 x 5 z 10 ( z x 4 ) 12 ( x + z ) 2 10 z 7 ( z x 4 ) 5

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x n x n + 3

x 2 n + 3

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a Δ a b b

a Δ + b +

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Exercises for review

( [link] ) What natural numbers can replace x so that the statement 5 < x 3 is true?

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( [link] ) Use the distributive property to expand 4 x ( 2 a + 3 b ) .

8 a x + 12 b x

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( [link] ) Express x x x y y y y ( a + b ) ( a + b ) using exponents.

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( [link] ) Find the value of 4 2 + 3 2 2 3 10 8 .

8

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( [link] ) Find the value of 4 2 + ( 3 + 2 ) 2 1 2 3 5 + 2 4 ( 3 2 2 3 ) 4 2 .

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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, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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