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Simplify: y −7 .

1 y 7

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Simplify: z −8 .

1 z 8

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When there is a product and an exponent we have to be careful to apply the exponent to the correct quantity. According to the order of operations , expressions in parentheses are simplified before exponents are applied. We’ll see how this works in the next example.

Simplify:

  1. 5 y −1
  2. ( 5 y ) −1
  3. ( −5 y ) −1

Solution

Notice the exponent applies to just the base y . 5 y −1
Take the reciprocal of y and change the sign of the exponent. 5 · 1 y 1
Simplify. 5 y
Here the parentheses make the exponent apply to the base 5 y . ( 5 y ) −1
Take the reciprocal of 5 y and change the sign of the exponent. 1 ( 5 y ) 1
Simplify. 1 5 y
( −5 y ) −1
The base is 5 y . Take the reciprocal of 5 y and change the sign of the exponent. 1 ( −5 y ) 1
Simplify. 1 −5 y
Use a b = a b . 1 5 y
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Simplify:

  1. 8 p −1
  2. ( 8 p ) −1
  3. ( −8 p ) −1

  1. 8 p
  2. 1 8 p
  3. 1 8 p

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Simplify:

  1. 11 q −1
  2. ( 11 q ) −1
  3. ( −11 q ) −1

  1. 11 q
  2. 1 11 q
  3. 1 11 q

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Now that we have defined negative exponents, the Quotient Property of Exponents needs only one form, a m a n = a m n , where a 0 and m and n are integers.

When the exponent in the denominator is larger than the exponent in the numerator, the exponent of the quotient will be negative. If the result gives us a negative exponent, we will rewrite it by using the definition of negative exponents, a n = 1 a n .

Simplify expressions with integer exponents

All the exponent properties we developed earlier in this chapter with whole number exponents apply to integer exponents, too. We restate them here for reference.

Summary of exponent properties

If a , b are real numbers and m , n are integers, then

Product Property a m · a n = a m + n Power Property ( a m ) n = a m · n Product to a Power Property ( a b ) m = a m b m Quotient Property a m a n = a m n , a 0 Zero Exponent Property a 0 = 1 , a 0 Quotient to a Power Property ( a b ) m = a m b m , b 0 Definition of Negative Exponent a n = 1 a n

Simplify:

  1. x −4 · x 6
  2. y −6 · y 4
  3. z −5 · z −3

Solution

x −4 · x 6
Use the Product Property, a m · a n = a m + n . x −4 + 6
Simplify. x 2
y −6 · y 4
The bases are the same, so add the exponents. y −6 + 4
Simplify. y −2
Use the definition of a negative exponent, a n = 1 a n . 1 y 2
z −5 · z −3
The bases are the same, so add the exponents. z −5−3
Simplify. z −8
Use the definition of a negative exponent, a n = 1 a n . 1 z 8
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Simplify:

  1. x −3 · x 7
  2. y −7 · y 2
  3. z −4 · z −5

  1. x 4
  2. 1 y 5
  3. 1 z 9

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Simplify:

  1. a −1 · a 6
  2. b −8 · b 4
  3. c −8 · c −7

  1. a 5
  2. 1 b 4
  3. 1 c 15

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In the next two examples, we’ll start by using the Commutative Property to group the same variables together. This makes it easier to identify the like bases before using the Product Property of Exponents .

Simplify: ( m 4 n −3 ) ( m −5 n −2 ) .

Solution

( m 4 n −3 ) ( m −5 n −2 )
Use the Commutative Property to get like bases together. m 4 m −5 · n −2 n −3
Add the exponents for each base. m −1 · n −5
Take reciprocals and change the signs of the exponents. 1 m 1 · 1 n 5
Simplify. 1 m n 5
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Simplify: ( p 6 q −2 ) ( p −9 q −1 ) .

1 p 3 q 3

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Simplify: ( r 5 s −3 ) ( r −7 s −5 ) .

1 r 2 s 8

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If the monomials have numerical coefficients, we multiply the coefficients, just as we did in Integer Exponents and Scientific Notation .

Simplify: ( 2 x −6 y 8 ) ( −5 x 5 y −3 ) .

Solution

( 2 x −6 y 8 ) ( −5 x 5 y −3 )
Rewrite with the like bases together. 2 ( −5 ) · ( x −6 x 5 ) · ( y 8 y −3 )
Simplify. −10 · x −1 · y 5
Use the definition of a negative exponent, a n = 1 a n . −10 · 1 x 1 · y 5
Simplify. −10 y 5 x
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Simplify: ( 3 u −5 v 7 ) ( −4 u 4 v −2 ) .

12 v 5 u

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Simplify: ( −6 c −6 d 4 ) ( −5 c −2 d −1 ) .

30 d 3 c 8

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Questions & Answers

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
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a perfect square v²+2v+_
Dearan Reply
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Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
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yes
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I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
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
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
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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
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.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
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how do you translate this in Algebraic Expressions
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Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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Practice Key Terms 2

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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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