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Introduction

You should know by now what the nthnth root of a number means. If the nthnth root of a number cannot be simplified to a rational number, we call it a surdsurd. For example, 22 and 6363 are surds, but 44 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 anan, where aa is any positive number, for example 77 or 5353. It is very common for nn to be 2, so we usually do not write a2a2. Instead we write the surd as just aa, 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 33 is on the number line. So how do we know where surds lie on the number line? From a calculator we know that 33 is equal to 1,73205...1,73205.... It is easy to see that 33 is above 1 and below 2. But to see this for other surds like 1818 without using a calculator, you must first understand the following fact:

Note: Interesting Fact :

If aa and bb are positive whole numbers, and a<ba<b, then an<bnan<bn. (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 1818 is? Well, you can easily see that 18<2518<25. Using our rule, we also know that 18<2518<25. But we know that 52=2552=25 so that 25=525=5. Now it is easy to simplify to get 18<518<5. Now we have a better idea of what 1818 is.

Now we know that 1818 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<1816<18. Using our rule again, we also know that 16<1816<18. But we know that 16 is a perfect square, so we can simplify 1616 to 4, and so we get 4<184<18!

As you can see, we have shown that 1818 is between 4 and 5. If we check on our calculator, we can see that 18=4,1231...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 42=1642=16, and the smallest perfect square greater than 18 was 52=2552=25. Here is a quick summary of what a perfect square or cube is:

Note: Interesting Fact :

A perfect square is the number obtained when an integer is squared. For example, 9 is a perfect square since 32=932=9. Similarly, a perfect cube is a number which is the cube of an integer. For example, 27 is a perfect cube, because 33=2733=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 Table 1.

Table 1: 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 523523 you should be able to tell that it lies somewhere between 3 and 4, because 273=3273=3 and 643=4643=4 and 52 is between 27 and 64. In fact 523=3,73...523=3,73... which is indeed between 3 and 4.

Exercise 1: Estimating Surds

Find the two consecutive integers such that 2626 lies between them.

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

Solution

  1. Step 1. From the table find the largest perfect square below 26 :

    This is 52=2552=25. Therefore 5<265<26.

  2. Step 2. From the table find the smallest perfect square above 26 :

    This is 62=3662=36. Therefore 26<626<6.

  3. Step 3. Put the inequalities together :

    Our answer is 5<26<65<26<6.

Exercise 2: Estimating Surds

493493 lies between:

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

Solution

  1. Step 1. Consider (a) as the solution :

    If 1<493<21<493<2 then cubing all terms gives 1<49<231<49<23. Simplifying gives 1<49<81<49<8 which is false. So 493493 does not lie between 1 and 2.

  2. Step 2. Consider (b) as the solution :

    If 2<493<32<493<3 then cubing all terms gives 23<49<3323<49<33. Simplifying gives 8<49<278<49<27 which is false. So 493493 does not lie between 2 and 3.

  3. Step 3. Consider (c) as the solution :

    If 3<493<43<493<4 then cubing all terms gives 33<49<4333<49<43. Simplifying gives 27<49<6427<49<64 which is true. So 493493 lies between 3 and 4.

Summary

  • If the nthnth root of a number cannot be simplified to a rational number, we call it a surdsurd
  • If aa and bb are positive whole numbers, and a<ba<b, then an<bnan<bn
  • 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. 55 lies between:
      1. 1 and 2
      2. 2 and 3
      3. 3 and 4
      4. 4 and 5
      Click here for the solution
    2. 1010 lies between:
      1. 1 and 2
      2. 2 and 3
      3. 3 and 4
      4. 4 and 5
      Click here for the solution
    3. 2020 lies between:
      1. 2 and 3
      2. 3 and 4
      3. 4 and 5
      4. 5 and 6
      Click here for the solution
    4. 3030 lies between:
      1. 3 and 4
      2. 4 and 5
      3. 5 and 6
      4. 6 and 7
      Click here for the solution
    5. 5353 lies between:
      1. 1 and 2
      2. 2 and 3
      3. 3 and 4
      4. 4 and 5
      Click here for the solution
    6. 103103 lies between:
      1. 1 and 2
      2. 2 and 3
      3. 3 and 4
      4. 4 and 5
      Click here for the solution
    7. 203203 lies between:
      1. 2 and 3
      2. 3 and 4
      3. 4 and 5
      4. 5 and 6
      Click here for the solution
    8. 303303 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 77 lies between them. Click here for the solution
  3. Find two consecutive integers such that 1515 lies between them. Click here for the solution

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