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Surd calculations

There are several laws that make working with surds (or roots) easier. We will list them all and then explain where each rule comes from in detail.

a n b n = a b n a b n = a n b n a m n = a m n

Surd law 1: a n b n = a b n

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, a n = a 1 n and b n = b 1 n . Then,

a n b n = a 1 n b 1 n = ( a b ) 1 n = a b n

Some examples using this law:

  1. 16 3 × 4 3 = 64 3 = 4
  2. 2 × 32 = 64 = 8
  3. a 2 b 3 × b 5 c 4 = a 2 b 8 c 4 = a b 4 c 2

Surd law 2: a b n = a n b n

If we look at a b n in exponential notation and apply the exponential laws then,

a b n = a b 1 n = a 1 n b 1 n = a n b n

Some examples using this law:

  1. 12 ÷ 3 = 4 = 2
  2. 24 3 ÷ 3 3 = 8 3 = 2
  3. a 2 b 13 ÷ b 5 = a 2 b 8 = a b 4

Surd law 3: a m n = a m n

If we look at a m n in exponential notation and apply the exponential laws then,

a m n = ( a m ) 1 n = a m n

For example,

2 3 6 = 2 3 6 = 2 1 2 = 2

Like and unlike surds

Two surds a m and b n are called like surds if m = n , otherwise they are called unlike surds . For example 2 and 3 are like surds, however 2 and 2 3 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

a m n = a b m b n

to rewrite the unlike surds so that b n is the same for all the surds.

Simplify to like surds as far as possible, showing all steps: 3 3 × 5 5

  1. = 3 5 15 × 5 3 15
  2. = 3 5 . 5 3 15 = 243 × 125 15 = 30 375 15

Simplest surd form

In most cases, when working with surds, answers are given in simplest surd form. For example,

50 = 25 × 2 = 25 × 2 = 5 2

5 2 is the simplest surd form of 50 .

Rewrite 18 in the simplest surd form:

  1. 18 = 2 × 9 = 2 × 3 × 3 = 2 × 3 × 3 = 2 × 3 2 = 3 2

Simplify: 147 + 108

  1. 147 + 108 = 49 × 3 + 36 × 3 = 7 2 × 3 + 6 2 × 3
  2. = 7 3 + 6 3
  3. = 13 3

This video gives some examples of simplifying surds.

Khan academy video on surds - 1

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 a + b (where a and b are rational) can be changed into a rational number by multiplying by a - b (similarly a - b can be rationalised by multiplying by a + b ). This is because

( a + b ) ( a - b ) = a - b

which is rational (since a and b are rational).

If we have a fraction which has a denominator which looks like a + b , then we can simply multiply both top and bottom by a - b achieving a rational denominator.

c a + b = a - b a - b × c a + b = c a - c b a - b

or similarly

c a - b = a + b a + b × c a - b = c a + c b a - b

Rationalise the denominator of: 5 x - 16 x

  1. To get rid of x in the denominator, you can multiply it out by another x . This "rationalises" the surd in the denominator. Note that x x = 1, thus the equation becomes rationalised by multiplying by 1 and thus still says the same thing.

    5 x - 16 x × x x
  2. The surd is expressed in the numerator which is the prefered way to write expressions. (That's why denominators get rationalised.)

    5 x x - 16 x x = ( x ) ( 5 x - 16 ) x

Rationalise the following: 5 x - 16 y - 10

  1. 5 x - 16 y - 10 × y + 10 y + 10
  2. 5 x y - 16 y + 50 x - 160 y - 100
  3. All the terms in the numerator are different and cannot be simplified and the denominator does not have any surds in it anymore.

Simplify the following: y - 25 y + 5

  1. y - 25 y + 5 × y - 5 y - 5
  2. y y - 25 y - 5 y + 125 y - 25 = y ( y - 25 ) - 5 ( y - 25 ) ( y - 25 ) = ( y - 25 ) ( y - 25 ) ( y - 25 ) = y - 25

The following video explains some of the concepts of rationalising the denominator.

Khan academy video on surds - 2

End of chapter exercises

  1. Expand:
    ( x - 2 ) ( x + 2 )
  2. Rationalise the denominator:
    10 x - 1 x
  3. Write as a single fraction:
    3 2 x + x
  4. Write in simplest surd form:
    1. 72
    2. 45 + 80
    3. 48 12
    4. 18 ÷ 72 8
    5. 4 ( 8 ÷ 2 )
    6. 16 ( 20 ÷ 12 )
  5. Expand and simplify:
    ( 2 + 2 ) 2
  6. Expand and simplify:
    ( 2 + 2 ) ( 1 + 8 )
  7. Expand and simplify:
    ( 1 + 3 ) ( 1 + 8 + 3 )
  8. Rationalise the denominator:
    y - 4 y - 2
  9. Rationalise the denominator:
    2 x - 20 y - 10
  10. Prove (without the use of a calculator) that:
    8 3 + 5 5 3 - 1 6 = 13 2 2 3
  11. Simplify, without use of a calculator:
    98 - 8 50
  12. Simplify, without use of a calculator:
    5 ( 45 + 2 80 )
  13. Write the following with a rational denominator:
    5 + 2 5
  14. Simplify:
    98 x 6 + 128 x 6
  15. Evaluate without using a calculator: 2 - 7 2 1 2 . 2 + 7 2 1 2
  16. The use of a calculator is not permissible in this question. Simplify completely by showing all your steps: 3 - 1 2 12 + ( 3 3 ) 3
  17. Fill in the blank surd-form number which will make the following equation a true statement: - 3 6 × - 2 24 = - 18 × ______

Questions & Answers

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
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
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
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
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Source:  OpenStax, Siyavula textbooks: grade 11 maths. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11243/1.3
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