Come On Lets Do Surds!

Surds have always troubled mathematicians and students alike. The square root of lets say 5, does look rather ugly, and that`s why we have to learn how to manipulate them to try to make them a little less unwieldy.

Rooting a square number is easy. 4, 25 and 225, nicely root to 2, 5 and 15 respectively. But what happens when we try to root 5 for example? We know the root must lie in between 2 and 3 but what exact number when timesed by itself will equal 5? Our calculator helpfully says, 2.2360679775. But the problem is that isn`t exact. The number is irrational and can never be found in its completed form. So if we times 2.236... by itself we do nearly get 5 but never exactly. This doesn`t make for neat answers.

This is why we have to learn how to tame the surd. lets take the square root of 20. 20 can be described as 4 x5. We notice immediately that 4 is in fact a square number it roots to 2, so already we can describe the root of 20 as 2 root 5. This is an improvement but still messy. But the point is to clean up the surd as much as possible. lets look at the square root of 180. Again 180 can be described as 36 x 5. 36 roots to 6, so 180 is actually 6 root 5. If we were to add the root of 20 to the root of 180 we can now do it. it is 2 root 5 plus 6 root 5, i.e. we have 8 root 5. Now we are getting somewhere. How about if we divide root 180 by root 20, we find we have 6 root 5 divided by 2 root 5. The two roots cancel each other out and we are left with 6 divided by 2, which equals 3. That is as neat as we want it!

Where surds can really cause problems in calculations is when they are the denominator of a sum. Take 4 divided by root 2. Because we are dividing by a non exact, or rather irrational number, the answer can run away from us, accuracy is compromised. So we manipulate the fraction. If we multiply the top and bottom by root 2 we get 4 root 2 divided by 2 and that equals 2 root 2 which again is much neater.

We can also do this with more complicated denominators. Say we have a denominator of root 2 plus 3 and it is dividing into 4. If we multiply by root 2 minus 3 on the top and bottom we are able to remove the root from the denominator. The top becomes 4 root 2 minus 9 divided by minus 7. Not exactly a neat fraction but much more easy to work with.