Surds Denominator

Surds are nothing but natural numbers which are written in square root form or cube root form or so on. And hence in other terms surds are irrational numbers. Surds Denominator is now defined as the denominator part of a fraction where the fraction consists a surd in the denominator of the fraction.

Having problem with Rationalize a Denominator keep reading my upcoming posts, i will try to help you.

Examples for Surds Denominator are as below.

`(1)/(sqrt(5))`

1 + 3
1 - √5

Rationalising the Surds Denominator

Lets see the rules to rationalise the surds denominators:

It is a bit uneasy to have a fraction having a surd denominator. For such fractions it is best to rationalize the Surd Denominator. Rationalising the Surd Denominator means multiply the numerator and denominator of the fraction by a specific expression. In this process the irrational form of the surd changes to rational form. Sometimes Surds Denominator occurs in complex form that is having complex numbers along with it. Complex Surds Denominator representation can be shown with the following example.

`(1-i sqrt(3))/(1+ i sqrt(3))`

Now multiply numerator and denominator by 1-i√3 , this gives in the denominator 1+32  = 10  and in the  numerator we will multiply 1-i√3 by 1-i√3 and that gives  4 -i2√3 . This is how we will solve complex Surds denominator.

This type of Surds Denominator can be rationalised by multiplying and dividing by the complex conjugate of the Surds Denominator.

Examples for rationalising the Surds Denominator

Let us consider the following 2 examples to know how to rationalise the Surds Denominator.

To rationalise the denominator of:

i)   `(1)/(sqrt(5))` .

For this type of Surds Denominator multiply the numerator and denominator of the fraction by √5. The numerator will become

√5 and the denominator will become 5 (√5 times √5 = 5).

`(1)/(sqrt(5))` * `(sqrt(5))/(sqrt(5))` = `(sqrt(5))/(5)`

ii) `(1+3)/(1- sqrt(5))`

For this type of Surds Denominator look at the denominator of the fraction and change the sign used inbetween as below. And then multiply the numerator and denominator of the fraction by this expression i.e.,

`(1+3)/(1- sqrt(5))`

= `(4)/(1- sqrt(5))` * `(1 + sqrt(5))/(1+ sqrt(5))`

= `(4+ 4 sqrt(5))/(1^2 - sqrt(5)^2)`    =  `(4 + 4sqrt(5))/(-4)` = -1 -`sqrt(5)`

This is how rationalising the Surds Denominator is done.