It is often useful to look at a surd in exponential notation as it allows us to use the exponential laws we learnt in Grade 10. In exponential notation,
$\sqrt[n]{a}={a}^{\frac{1}{n}}$ and
$\sqrt[n]{b}={b}^{\frac{1}{n}}$ . Then,
Two surds
$\sqrt[m]{a}$ and
$\sqrt[n]{b}$ are called
like surds if
$m=n$ , otherwise they are called
unlike surds . For example
$\sqrt{2}$ and
$\sqrt{3}$ are like surds, however
$\sqrt{2}$ and
$\sqrt[3]{2}$ are unlike surds. An important thing to realise about the surd laws we have just learnt is that the surds in the laws are all like surds.
If we wish to use the surd laws on unlike surds, then we must first convert
them into like surds. In order to do this we use the formula
$$\sqrt[n]{{a}^{m}}=\sqrt[bn]{{a}^{bm}}$$
to rewrite the unlike surds so that
$bn$ is the same for all the surds.
Simplify to like surds as far as possible, showing all steps:
$\sqrt[3]{3}\times \sqrt[5]{5}$
This video gives some examples of simplifying surds.
Rationalising denominators
It is useful to work with fractions, which have rational denominators instead of surd denominators. It is possible to rewrite any fraction, which has a surd in the denominator as a fraction which has a rational denominator. We will now see how this can be achieved.
Any expression of the form
$\sqrt{a}+\sqrt{b}$ (where
$a$ and
$b$ are rational)
can be changed into a rational number by multiplying by
$\sqrt{a}-\sqrt{b}$ (similarly
$\sqrt{a}-\sqrt{b}$ can be rationalised by multiplying by
$\sqrt{a}+\sqrt{b}$ ). This is because
$$(\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b})=a-b$$
which is rational (since
$a$ and
$b$ are rational).
If we have a fraction which has a denominator which looks like
$\sqrt{a}+\sqrt{b}$ , then we can simply multiply both top and bottom by
$\sqrt{a}-\sqrt{b}$ achieving a rational denominator.
Rationalise the denominator of:
$\frac{5x-16}{\sqrt{x}}$
To get rid of
$\sqrt{x}$ in the denominator, you can multiply it out by another
$\sqrt{x}$ . This "rationalises" the surd in the denominator. Note that
$\frac{\sqrt{x}}{\sqrt{x}}$ = 1, thus the equation becomes rationalised by multiplying by 1 and thus still says the same thing.
Evaluate without using a calculator:
${\left(2,-,{\displaystyle \frac{\sqrt{7}}{2}}\right)}^{{\textstyle \frac{1}{2}}}\phantom{\rule{0.222222em}{0ex}}.\phantom{\rule{0.222222em}{0ex}}{\left(2,+,{\displaystyle \frac{\sqrt{7}}{2}}\right)}^{{\textstyle \frac{1}{2}}}$
The use of a calculator is not permissible in this question. Simplify completely by showing all your steps:
${3}^{-{\textstyle \frac{1}{2}}}\left[\sqrt{12},+,\sqrt[3]{\left(3\sqrt{3}\right)}\right]$
Fill in the blank surd-form number which will make the following equation a true statement:
$-3\sqrt{6}\times -2\sqrt{24}=-\sqrt{18}\times \mathrm{\_\_\_\_\_\_}$
Questions & Answers
anyone know any internet site where one can find nanotechnology papers?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
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.
Harper
Do you know which machine is used to that process?