Manipulation of Surds

Simplifying Surds

Consider

This means "the number which, when multiplied by itself, equals 50"

This must be the same as because this would mean "The number which, when multiplied by itself, comes to." (Which is obviously exactly the same).

Since, we can see that can be simplified to

In other words,

 

Addition and Subtraction

Adding and subtracting surds is relatively straightforward.

E.g.

 

Multiplication

Consider

This is performed as usual:

It is worth noting that which is rational

E.g. Find an expression which, when multiplied with will produce a rational product.

Using the bracket, we get

Division

We use the rule mentioned above to rationalise the denominator

E.g. Divide

We remove the surds from the bottom:

Note the bracket used to rationalise the bottom must also multiply the top.