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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 .

49 3 lies between:

(a) 1 and 2 (b) 2 and 3 (c) 3 and 4 (d) 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.

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. 5 lies between (a) 1 and 2 (b) 2 and 3 (c) 3 and 4 (d) 4 and 5 Click here for the solution
2. 10 lies between (a) 1 and 2 (b) 2 and 3 (c) 3 and 4 (d) 4 and 5 Click here for the solution
3. 20 lies between (a) 2 and 3 (b) 3 and 4 (c) 4 and 5 (d) 5 and 6 Click here for the solution
4. 30 lies between (a) 3 and 4 (b) 4 and 5 (c) 5 and 6 (d) 6 and 7 Click here for the solution
5. 5 3 lies between (a) 1 and 2 (b) 2 and 3 (c) 3 and 4 (d) 4 and 5 Click here for the solution
6. 10 3 lies between (a) 1 and 2 (b) 2 and 3 (c) 3 and 4 (d) 4 and 5 Click here for the solution
7. 20 3 lies between (a) 2 and 3 (b) 3 and 4 (c) 4 and 5 (d) 5 and 6 Click here for the solution
8. 30 3 lies between (a) 3 and 4 (b) 4 and 5 (c) 5 and 6 (d) 6 and 7 Click here for the solution
  1. Find two consecutive integers such that 7 lies between them. Click here for the solution
  2. Find two consecutive integers such that 15 lies between them. Click here for the solution

Questions & Answers

Application of nanotechnology in medicine
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
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What is meant by 'nano scale'?
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What is STMs full form?
LITNING
scanning tunneling microscope
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how nano science is used for hydrophobicity
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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
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
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
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Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
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why?
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what school?
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biomolecules are e building blocks of every organics and inorganic materials.
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anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
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Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
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Source:  OpenStax, Siyavula textbooks: grade 10 maths [ncs]. OpenStax CNX. Aug 05, 2011 Download for free at http://cnx.org/content/col11239/1.2
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