Processing math: 100%

Rationalizing the denominator

What is “Rationalizing the denominator”?


Certain fractions have radicals in their denominators. Some of these radicals can get simplified, but some do not get simplified like√2, because they are irrational numbers, or not perfect squares.

Since division by irrational numbers or radicals that cannot be simplified does not lead to a definite result, it is not considered proper to write a fraction with a radical in the denominator.

Thus if the radicals in the denominator of a fraction do not get simplified, we rationalize the denominator of the fraction, which means that we convert the irrational number/expression into a rational number/expression by means of mathematical operations.

How to rationalize the denominator of a fraction?

In order to rationalize the denominator of a fraction, you need to first determine whether you need to do conjugation, or simply multiply again and again.

If the denominator is a binomial, that means, if it contains two terms separated by a plus or a minus sign, then you have to perform conjugation in order to rationalize it.

On the other hand, if the denominator is just a radical, then all you need to do is multiply that radical again and again with the fraction as many times as is the power of the radical.

Denominator is a binomial

Example:

x+33+2

In this rational expression, the denominator is a binomial because it contains two terms separated by addition or subtraction. First find its conjugate as described here. Conjugate of 3+2 is 3-2 (simply change the sign on the second term).

Now multiply the conjugate with the numerator and denominator of the fraction.

x+33+23-23-2

Now you need to expand the denominator by FOIL as follows:

(x+3)(3-2)(3+2)(3-2)=(x+3)(3-2)9-32+32-22=(x+3)(3-2)9-2=(x+3)(3-2)7

Thus you have a fraction in which the denominator does not contain an irrational number or a radical:

x+33+2=(x+3)(3-2)7

The denominator is a radical


In order to rational a denominator which is a single term, multiply the denominator with the numerator and nominator of the fraction. For example,

2222=2(2)(2)(2)=2(2)2=2

If the denominator contains a radical that has a power greater than a square root, for instance, a cube root, you need to multiply it a total of three times as follows:

222222=2(2)(2)(2)(2)(2)=2(22)2=242=4

The denominator is π

If there is an irrational number like π in the denominator, you can’t rationalize the denominator even if you multiply by pi in the numerator and denominator because pi is a transcendental irrational number.

1 comment:

  1. Superb website you have here but I wwas curious if you knew of any discussion boards that cover the same topics talke about
    here? I'd really like to be a part of online community where I can get comments from other experienced people
    that share the same interest. If you hafe any recommendations, please let mee know.
    Kudos!

    ReplyDelete

Search This Blog