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 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:

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 7.1.

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.