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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 + 3)/(3 + √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 + 3)/(3 + √2)*(3- √2)/(3- √2)`

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

`((x + 3)(3- √2))/((3 + √2)(3- √2))=((x + 3)(3- √2))/(9-3√2+3√2-√2 √2)= ((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 + 3)/(3 + √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,

`2/(√2)*(√2)/(√2)=(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:

`2/(∛2)*(∛2)/(∛2)*(∛2)/(∛2)=(2*(∛2)*(∛2))/((∛2)*(∛2)*(∛2))=(2∛(2*2))/2=2* ∛(4)/2= ∛4`

The denominator is `pi`

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.

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