First of all we must remember what a surd is... A surd is any number which has a square root sign, but doesn't actually have a square root. So example's of surds are; √15, √30 and √47.

When we are trying to rationalise the denominator of a surd what that means is we want to get rid of any square root signs in the denominator. In maths it is very difficult to work with square roots in our denominator.

So to begin we must find a way to get the rid of the surd in our denominator in our example below:

√2  

√7

To acheive this what all we must do is to find the number that will get rid of the square root sign for us, which in our example is √7. So now we must multiply both the numerator and the denominator of our fraction by √7, because as you know if we do anything to one part of a fraction we must do it to the other. 

√2  x √7

√7    √7      

Which gives us the following answer;

√2  x √7 = (2 x 7) =  √14

√7    √7     √ (7 x 7)       7

Let's try one more example now;

 6√3

6√12 
If we are asked to rationalise the denominator of a surd for the above example we just follow the same steps. First of all we must try to get rid of the square root sign in our denominator and we can do this by multiplying it by whatever number is within the
√ sign.

 6√3 x √12 = 6√36

6√12   √12    6x12
Now we must just simply continue to work out our sums in our numerator and denominator, which leaves us with;

 6√3 x √12 = 6√36 = 6x6 = 36

6√12   √12     6x12     72     72

We have now succesfully rationalised the denominator of a surd, which leaves us with our final answer of 36.
                                  
72

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