Grade 10 Math by High School Science, Rory Adams, et al - HTML preview

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Estimating Surds

1. 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, m31339.id265290.png and m31339.id265299.png are surds, but m31339.id265311.png 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 m31339.id265327.png, where a is any positive number, for example m31339.id265352.png or m31339.id265361.png. It is very common for n to be 2, so we usually do not write m31339.id265387.png. Instead we write the surd as just m31339.id265398.png, 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 m31339.id265414.png is on the number line. So how do we know where surds lie on the number line? From a calculator we know that m31339.id265424.png is equal to 1,73205.... It is easy to see that m31339.id265441.png is above 1 and below 2. But to see this for other surds like m31339.id265451.png 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 m31339.id265698.png. (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 m31339.id265728.png is? Well, you can easily see that 18 < 25. Using our rule, we also know that m31339.id265749.png. But we know that 52 = 25 so that m31339.id265783.png. Now it is easy to simplify to get m31339.id265797.png. Now we have a better idea of what m31339.id265812.png is.

Now we know that m31339.id265826.png 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 m31339.id265851.png. But we know that 16 is a perfect square, so we can simplify m31339.id265866.png to 4, and so we get m31339.id265875.png!

As you can see, we have shown that m31339.id265896.png is between 4 and 5. If we check on our calculator, we can see that m31339.id265905.png, 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 = 16, and the smallest perfect square greater than 18 was 52 = 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 32 = 9. Similarly, a perfect cube is a number which is the cube of an integer. For example, 27 is a perfect cube, because 33 = 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 11.

Table 1. Some perfect squares and perfect cubes
IntegerPerfect SquarePerfect Cube
000
111
248
3927
41664
525125
636216
749343
864512
981729
101001000

When given the surd m31339.id266203.png you should be able to tell that it lies somewhere between 3 and 4, because m31339.id266214.png and m31339.id266231.png and 52 is between 27 and 64. In fact m31339.id266248.png which is indeed between 3 and 4.

Exercise 1. Estimating Surds (Go to Solution)

Find the two consecutive integers such that m31339.id266288.png lies between them.

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


Exercise 2. Estimating Surds (Go to Solution)

m31339.id266459.png lies between:

Table 2.
(a) 1 and 2(b) 2 and 3(c) 3 and 4(d) 4 and 5

2. 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 m31339.id266882.png

  • 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.

3. End of Chapter Exercises

Table 3.
1. m31339.id266966.png lies between(a) 1 and 2(b) 2 and 3(c) 3 and 4(d) 4 and 5 Click here for the solution
2. m31339.id267009.png lies between(a) 1 and 2(b) 2 and 3(c) 3 and 4(d) 4 and 5 Click here for the solution
3. m31339.id267051.png lies between(a) 2 and 3(b) 3 and 4(c) 4 and 5(d) 5 and 6 Click here for the solution
4. m31339.id267094.png lies between(a) 3 and 4(b) 4 and 5(c) 5 and 6(d) 6 and 7 Click here for the solution
5. m31339.id267137.png lies between(a) 1 and 2(b) 2 and 3(c) 3 and 4(d) 4 and 5 Click here for the solution
6. m31339.id267181.png lies between(a) 1 and 2(b) 2 and 3(c) 3 and 4(d) 4 and 5 Click here for the solution
7. m31339.id267226.png lies between(a) 2 and 3(b) 3 and 4(c) 4 and 5(d) 5 and 6 Click here for the solution
8. m31339.id267271.png 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 m31339.id267324.png lies between them. Click here for the solution

  2. Find two consecutive integers such that m31339.id267347.png lies between them. Click here for the solution

4. Solutions to Exercises

Solution to Exercise 1. (Return to Exercise)

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

    This is 52 = 25. Therefore m31339.id266348.png.

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

    This is 62 = 36. Therefore m31339.id266394.png.

  3. Put the inequalities together :

    Our answer is m31339.id266420.png.


Solution to Exercise 2. (Return to Exercise)

  1. Consider (a) as the solution :

    If m31339.id266531.png then cubing all terms gives 1 < 49 < 23. Simplifying gives 1 < 49 < 8 which is false. So m31339.id266593.png does not lie between 1 and 2.

  2. Consider (b) as the solution :

    If m31339.id266616.png then cubing all terms gives 23 < 49 < 33. Simplifying gives 8 < 49 < 27 which is false. So m31339.id266682.png does not lie between 2 and 3.

  3. Consider (c) as the solution :

    If m31339.id266705.png then cubing all terms gives 33 < 49 < 43. Simplifying gives 27 < 49 < 64 which is true. So m3 </div> </div> <div class=

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