Surds
Surds is just a fancy name for equations involving square roots. For example:
Now, the symbol over the top (called a radical symbol) might look scary at first, but don't panic! They aren't as complicated as you might at first think. There are just a few rules you need to know, and surds will be easy,
Now, the first of these rules is simple:
I know, I know, it seems obvious! But stay with me! This rule combined with the next lets us answer our first example question!
That rule is:
That rule is:
This is less obvious, but if we combine this rule with the first we can find a useful rule that allows us to simplify square roots.
Let's consider the following equation:
Let's consider the following equation:
Now, we can choose to simplify this in two ways: using either of the rules outlined above. We could use the first rule and cancel out the b squared into just b.
Or, we can use the second rule and combine the two square roots into one:
Both of these are correct, however the first is considered to be the simplest form. So, we can combine the two equations we have just found and find the rule which lets us simplify surds:
Okay, so what's my point?
Well, this means that if we have a number inside a radical, and it is a square number (b squared) multiplied by another number (a), we can square root b squared and take it outside the radical. Putting b outside the radical and leaving a unchanged.
So, if we were asked to simplify this:
Well, this means that if we have a number inside a radical, and it is a square number (b squared) multiplied by another number (a), we can square root b squared and take it outside the radical. Putting b outside the radical and leaving a unchanged.
So, if we were asked to simplify this:
We can use our rule to simplify it like so:
The third step I added for clarity to show what's really going on when you "Take 36 out from the radical". However in an exam you can leave that step out. It will save you time on the questions to get used to missing out the third step, it isn't that difficult to do. Here are some examples with other numbers:
And so on and so forth.
Now, what about fractions combined with surds? Sounds even more painful, doesn't it? Well, fear not! It's no harder than fractions alone. Well, that's a lie... It's not a lot harder than fractions alone.
There are two kinds of questions regarding fractional surds. To tackle the first kind you need to know one more rule:
Now, what about fractions combined with surds? Sounds even more painful, doesn't it? Well, fear not! It's no harder than fractions alone. Well, that's a lie... It's not a lot harder than fractions alone.
There are two kinds of questions regarding fractional surds. To tackle the first kind you need to know one more rule:
So, here's a simple example question:
Simplify this:
Simplify this:
So, using the new rule:
Done! Easy, right?
Now, some examples:
Now, some examples:
Now, the answer to the final example would not be allowed in the exam. As a general rule: radicals should never be on the bottom of a fraction. In the final case, you should multiply the top and bottom of the fraction by the square root of 5, the value of the fraction is unchanged as you have multiplied the top and the bottom by the same number, however you will remove the square root. As I will demonstrate:
Now, this leads to our final type of question. The final rule about not leaving radicals in the denominator can cause a few problems when you have more complicated denominators. for example:
In this example you can't easily cancel the radical out of the denominator as before. So, how do we get rid of the radical? Well, we must use something called the difference of two squares, or D.O.T.S. D.O.T.S is a way of factorising a statement which contains a square number, minus another square number. It looks like this:
Now, our denominator is of the form a+b. If we multiply the top and bottom by a-b the radical should cancel. Let's try it:
So, it works. You should note that the fraction won't disappear in all cases, but the radical always will.