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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, and
are surds, but
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 , where
a
is any positive number, for example
or
. It is very common for
n
to be 2, so we usually do not write
. Instead we write the surd as just
, 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 is on the number line. So how do we know where surds lie on the number line? From a calculator we know that
is equal to 1,73205.... It is easy to see that
is above 1 and below 2. But to see this for other surds like
without using a calculator, you must first understand the following fact:
If
a
and
b
are positive whole numbers, and
a < b
, then . (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 is? Well, you can easily see that 18 < 25. Using our rule, we also know that
. But we know that 52 = 25 so that
. Now it is easy to simplify to get
. Now we have a better idea of what
is.
Now we know that 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
. But we know that 16 is a perfect square, so we can simplify
to 4, and so we get
!
As you can see, we have shown that is between 4 and 5. If we check on our calculator, we can see that
, 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:
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.
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 you should be able to tell that it lies somewhere between 3 and 4, because
and
and 52 is between 27 and 64. In fact
which is indeed between 3 and 4.
Exercise 1. Estimating Surds (Go to Solution)
Find the two consecutive integers such that 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)
lies between:
(a) 1 and 2 | (b) 2 and 3 | (c) 3 and 4 | (d) 4 and 5 |
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
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.
1. |
![]() | (a) 1 and 2 | (b) 2 and 3 | (c) 3 and 4 | (d) 4 and 5 | Click here for the solution |
2. |
![]() | (a) 1 and 2 | (b) 2 and 3 | (c) 3 and 4 | (d) 4 and 5 | Click here for the solution |
3. |
![]() | (a) 2 and 3 | (b) 3 and 4 | (c) 4 and 5 | (d) 5 and 6 | Click here for the solution |
4. |
![]() | (a) 3 and 4 | (b) 4 and 5 | (c) 5 and 6 | (d) 6 and 7 | Click here for the solution |
5. |
![]() | (a) 1 and 2 | (b) 2 and 3 | (c) 3 and 4 | (d) 4 and 5 | Click here for the solution |
6. |
![]() | (a) 1 and 2 | (b) 2 and 3 | (c) 3 and 4 | (d) 4 and 5 | Click here for the solution |
7. |
![]() | (a) 2 and 3 | (b) 3 and 4 | (c) 4 and 5 | (d) 5 and 6 | Click here for the solution |
8. |
![]() | (a) 3 and 4 | (b) 4 and 5 | (c) 5 and 6 | (d) 6 and 7 | Click here for the solution |
Find two consecutive integers such that lies between them. Click here for the solution
Find two consecutive integers such that lies between them. Click here for the solution
Solution to Exercise 1. (Return to Exercise)
This is 52 = 25. Therefore .
This is 62 = 36. Therefore .
Our answer is .
Solution to Exercise 2. (Return to Exercise)
If then cubing all terms gives 1 < 49 < 23. Simplifying gives 1 < 49 < 8 which is false. So
does not lie between 1 and 2.
If then cubing all terms gives 23 < 49 < 33. Simplifying gives 8 < 49 < 27 which is false. So
does not lie between 2 and 3.
If then cubing all terms gives 33 < 49 < 43. Simplifying gives 27 < 49 < 64 which is true. So