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Introduction

You should know by now what the n th root of a number means. If the n th root of a number cannot be simplified to a rational number, we call it a surd . For example, 2 and 6 3 are surds, but 4 is not a surd because it can be simplified to the rational number 2.

In this chapter we will only look at surds that look like a n , where a is any positive number, for example 7 or 5 3 . It is very common for n to be 2, so we usually do not write a 2 . Instead we write the surd as just a , which is much easier to read.

It is sometimes useful to know the approximate value of a surd without having to use a calculator. For example, we want to be able to estimate where a surd like 3 is on the number line. So how do we know where surds lie on the number line? From a calculator we know that 3 is equal to 1 , 73205 . . . . It is easy to see that 3 is above 1 and below 2. But to see this for other surds like 18 without using a calculator, you must first understand the following fact:

Interesting fact

If a and b are positive whole numbers, and a < b , then a n < b n . (Challenge: Can you explain why?)

If you don't believe this fact, check it for a few numbers to convince yourself it is true.

How do we use this fact to help us guess what 18 is? Well, you can easily see that 18 < 25 . Using our rule, we also know that 18 < 25 . But we know that 5 2 = 25 so that 25 = 5 . Now it is easy to simplify to get 18 < 5 . Now we have a better idea of what 18 is.

Now we know that 18 is less than 5, but this is only half the story. We can use the same trick again, but this time with 18 on the right-hand side. You will agree that 16 < 18 . Using our rule again, we also know that 16 < 18 . But we know that 16 is a perfect square, so we can simplify 16 to 4, and so we get 4 < 18 !

As you can see, we have shown that 18 is between 4 and 5. If we check on our calculator, we can see that 18 = 4 , 1231 . . . , and the idea was right! You will notice that our idea used perfect squares that were close to the number 18. We found the largest perfect square smaller than 18 was 4 2 = 16 , and the smallest perfect square greater than 18 was 5 2 = 25 . Here is a quick summary of what a perfect square or cube is:

Interesting fact

A perfect square is the number obtained when an integer is squared. For example, 9 is a perfect square since 3 2 = 9 . Similarly, a perfect cube is a number which is the cube of an integer. For example, 27 is a perfect cube, because 3 3 = 27 .

To make it easier to use our idea, we will create a list of some of the perfect squares and perfect cubes. The list is shown in [link] .

Some perfect squares and perfect cubes
Integer Perfect Square Perfect Cube
0 0 0
1 1 1
2 4 8
3 9 27
4 16 64
5 25 125
6 36 216
7 49 343
8 64 512
9 81 729
10 100 1000

When given the surd 52 3 you should be able to tell that it lies somewhere between 3 and 4, because 27 3 = 3 and 64 3 = 4 and 52 is between 27 and 64. In fact 52 3 = 3 , 73 ... which is indeed between 3 and 4.

Find the two consecutive integers such that 26 lies between them.

(Remember that consecutive numbers are two numbers one after the other, like 5 and 6 or 8 and 9.)

  1. This is 5 2 = 25 . Therefore 5 < 26 .

  2. This is 6 2 = 36 . Therefore 26 < 6 .

  3. Our answer is 5 < 26 < 6 .

Got questions? Get instant answers now!

49 3 lies between:

  1. 1 and 2
  2. 2 and 3
  3. 3 and 4
  4. 4 and 5

  1. If 1 < 49 3 < 2 then cubing all terms gives 1 < 49 < 2 3 . Simplifying gives 1 < 49 < 8 which is false. So 49 3 does not lie between 1 and 2.

  2. If 2 < 49 3 < 3 then cubing all terms gives 2 3 < 49 < 3 3 . Simplifying gives 8 < 49 < 27 which is false. So 49 3 does not lie between 2 and 3.

  3. If 3 < 49 3 < 4 then cubing all terms gives 3 3 < 49 < 4 3 . Simplifying gives 27 < 49 < 64 which is true. So 49 3 lies between 3 and 4.

Got questions? Get instant answers now!

Summary

  • If the n th root of a number cannot be simplified to a rational number, we call it a surd
  • If a and b are positive whole numbers, and a < b , then a n < b n
  • Surds can be estimated by finding the largest perfect square (or perfect cube) that is less than the surd and the smallest perfect square (or perfect cube) that is greater than the surd. The surd lies between these two numbers.

End of chapter exercises

  1. Answer the following multiple choice questions:
    1. 5 lies between:
      1. 1 and 2
      2. 2 and 3
      3. 3 and 4
      4. 4 and 5
      Click here for the solution
    2. 10 lies between:
      1. 1 and 2
      2. 2 and 3
      3. 3 and 4
      4. 4 and 5
      Click here for the solution
    3. 20 lies between:
      1. 2 and 3
      2. 3 and 4
      3. 4 and 5
      4. 5 and 6
      Click here for the solution
    4. 30 lies between:
      1. 3 and 4
      2. 4 and 5
      3. 5 and 6
      4. 6 and 7
      Click here for the solution
    5. 5 3 lies between:
      1. 1 and 2
      2. 2 and 3
      3. 3 and 4
      4. 4 and 5
      Click here for the solution
    6. 10 3 lies between:
      1. 1 and 2
      2. 2 and 3
      3. 3 and 4
      4. 4 and 5
      Click here for the solution
    7. 20 3 lies between:
      1. 2 and 3
      2. 3 and 4
      3. 4 and 5
      4. 5 and 6
      Click here for the solution
    8. 30 3 lies between:
      1. 3 and 4
      2. 4 and 5
      3. 5 and 6
      4. 6 and 7
      Click here for the solution
  2. Find two consecutive integers such that 7 lies between them. Click here for the solution
  3. Find two consecutive integers such that 15 lies between them. Click here for the solution

Questions & Answers

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?
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nano basically means 10^(-9). nanometer is a unit to measure length.
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do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
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absolutely yes
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how to know photocatalytic properties of tio2 nanoparticles...what to do now
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it is a goid question and i want to know the answer as well
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characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
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
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
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Source:  OpenStax, Siyavula textbooks: grade 10 maths [caps]. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11306/1.4
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