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
what is variations in raman spectra for nanomaterials
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
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?