How to Rationalize the Denominator | A Step-by-Step Guide with Examples

rationalizing the denominator

Rationalizing the denominator is a technique used to eliminate the presence of radicals (square roots or cube roots, etc

Rationalizing the denominator is a technique used to eliminate the presence of radicals (square roots or cube roots, etc.) in the denominator of a fraction. The aim is to simplify the fraction and make it easier to work with.

To rationalize the denominator, you need to multiply the numerator and denominator of the fraction by a suitable expression so that the resulting denominator no longer contains any radicals. This is achieved by multiplying the fraction by a conjugate of the denominator.

A conjugate is formed by changing the sign between the terms of a binomial expression. For example, if you have a denominator √5, then the conjugate would be √5. Multiplying the fraction by its conjugate allows you to eliminate the root because when you multiply a term by its conjugate, the resulting expression becomes a difference of squares, which eliminates the radical.

Let’s take an example to understand this process better:

Consider the fraction 3 / √2. To rationalize the denominator, we multiply the fraction by the conjugate of the denominator.

Conjugate of √2 is √2. Therefore, to rationalize the denominator of 3 / √2, we multiply both the numerator and denominator by √2:

(3 / √2) * (√2 / √2)

This simplifies to:

(3√2) / (√2 * √2)

The denominator now becomes:

(3√2) / 2

This resulting fraction, (3√2) / 2, is the rationalized form of the original fraction 3 / √2. It has no radicals in the denominator, making it easier to work with and compare with other fractions.

Rationalizing the denominator is particularly useful when simplifying expressions, solving equations, or working with complex fractions. It allows for clearer calculations and easier manipulation of the numbers involved.

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