Application using exponential law 3:
${a}^{-n}=\frac{1}{{a}^{n}},a\ne 0$
${2}^{-2}=\frac{1}{{2}^{2}}$
$\frac{{2}^{-2}}{{3}^{2}}$
${\left(\frac{2}{3}\right)}^{-3}$
$\frac{m}{{n}^{-4}}$
$\frac{{a}^{-3}\xb7{x}^{4}}{{a}^{5}\xb7{x}^{-2}}$
Exponential law 4:
${a}^{m}\xf7{a}^{n}={a}^{m-n}$
We already realised with law 3 that a minus sign is another way of saying that the exponential number is to be divided instead of multiplied. Law 4 is just a more general way of saying the same thing. We get this law by multiplying law 3 by
${a}^{m}$ on both sides and using law 2.
Application using exponential law 5:
${\left(ab\right)}^{n}={a}^{n}{b}^{n}$
${\left(2xy\right)}^{3}={2}^{3}{x}^{3}{y}^{3}$
${\left(\frac{7a}{b}\right)}^{2}$
${\left(5a\right)}^{3}$
Exponential law 6:
${\left({a}^{m}\right)}^{n}={a}^{mn}$
We can find the exponential of an exponential of a number. An exponential of a number is just a real number. So, even though the sentence sounds complicated, it is just saying that you can find the exponential of a number and then take the exponential of that number. You just take the exponential twice, using the answer of the first exponential as the argument for the second one.
Match the answers to the questions, by filling in the correct answer into the
Answer column.
Possible answers are:
$\frac{3}{2}$ , 1,
$-1$ ,
$-\frac{1}{3}$ , 8. Answers may be repeated.
The following video gives an example on using some of the concepts covered in this chapter.
Summary
Exponential notation means a number written like
$${a}^{n}$$ where
$n$ is an integer and
$a$ can be any real number.
$a$ is called the
base and
$n$ is called the
exponent or
index .
The
${n}^{\mathrm{th}}$ power of
$a$ is defined as:
$${a}^{n}=a\times a\times \cdots \times a\phantom{\rule{2.em}{0ex}}\left(\mathrm{n\; times}\right)$$
There are six laws of exponents:
Exponential Law 1:
${a}^{0}=1$
Exponential Law 2:
${a}^{m}\times {a}^{n}={a}^{m+n}$
Exponential Law 3:
${a}^{-n}=\frac{1}{{a}^{n}},\phantom{\rule{1.em}{0ex}}a\ne 0$
Exponential Law 4:
${a}^{m}\xf7{a}^{n}={a}^{m-n}$
Exponential Law 5:
${\left(ab\right)}^{n}={a}^{n}{b}^{n}$
Exponential Law 6:
${\left({a}^{m}\right)}^{n}={a}^{mn}$
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.
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
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
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.