Rationalizing denominators why

Like so many things it is nothing to get particularly obsessed about, but knowing how to rationalise denominators is quite a useful tool to have at one's disposition. In fact more so in the general context of manipulating expressions than just for simplifying numbers. It is based on a small trick that is easy to understand, but which most people would probably not have thought of if it were not taught to them.

Also I think that a similar method though maybe better called realising than rationalising is used in the most striaghtforward proof of the fact that the complex numbers are a field. Rationalizing the denominator RTD a special case of the method of simpler multiples is useful because it often serves to simplify problems, e. This can lead to all sorts of simplifications, e. Here's another example from number theory showing how RTD serves to reduce divisibility of algebraic integers to rational integers.

Harold Edwards: Divisor Theory. We can also "rationalize" to base fields in any algebraic extension, e. The reason I've seen is historical, because the Greeks could easily construct a fraction with radicals on the numerator by constructing all the radicals and then dividing as necessary, but to divide by an irrational amount is very difficult. This problem is changed to simply dividing up rationals which is easily done with straightedge and compass.

Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Why rationalize the denominator? Ask Question. Asked 6 years, 10 months ago. Active 1 month ago. Viewed 29k times.

Ahaan S. Rungta 7, 5 5 gold badges 25 25 silver badges 63 63 bronze badges. Reinstate Monica Reinstate Monica 4, 2 2 gold badges 17 17 silver badges 34 34 bronze badges. Many answers and explanations there. The discussion is pedagogical rather than mathematical, but anyway. A quick summary: comparing answers is easier in standard form, b4 computers yadda yadda, nice to learn how to do this so that you can when it really matters.

What is Rationalizing? Rationalize the Denominator Using Conjugates 3. Rationalize the Denominator Using Algebraic Identities 4. Rationalize the Denominator With 3 Terms 5.

Rationalize the Denominator Using Conjugates. Rationalize the Denominator Using Algebraic Identities. Rationalize the Denominator With 3 Terms. Have questions on basic mathematical concepts?

Become a problem-solving champ using logic, not rules. Use the Distributive Property to multiply the binomials in the numerator and denominator. Since you multiplied by the conjugate of the denominator, the radical terms in the denominator will combine to 0.

Simplify radicals where possible. Then multiply the numerator and denominator by. One word of caution: this method will work for binomials that include a square root, but not for binomials with roots greater than 2.

This is because squaring a root that has an index greater than 2 does not remove the root, as shown below. There are no cubed numbers to pull out! Multiplying by its conjugate does not result in a radical-free expression.

Identify the conjugate of the denominator. The conjugate will be the binomial that, when multiplied by the denominator, eliminates the radical. The conjugate is. The denominator is not the conjugate. Look for the binomial that, when multiplied by the denominator, eliminates the radical.

Multiplying the denominator by will not remove the radical. Look for the binomial that follows the pattern. When you encounter a fraction that contains a radical in the denominator, you can eliminate the radical by using a process called rationalizing the denominator.

To rationalize a denominator, you need to find a quantity that, when multiplied by the denominator, will create a rational number no radical terms in the denominator. When the denominator contains a single term, as in , multiplying the fraction by will remove the radical from the denominator.

When the denominator contains two terms, as in , identify the conjugate of the denominator, here , and multiply both numerator and denominator by the conjugate. Example Problem Rationalize the denominator. Example Problem Rationalize the denominator and simplify. B Incorrect. C Incorrect. Rationalizing the Numerator with one term As mentioned above, when a radical cannot be evaluated, for example, the square root of 3 or cube root of 5, it is called an irrational number.

So, in order to rationalize the numerator, we need to get rid of all radicals that are in the numerator. Note that these are the same basic steps for rationalizing a denominator, we are just applying to the numerator now.

Step 1: Multiply numerator and denominator by a radical that will get rid of the radical in the numerator. If the radical in the numerator is a square root, then you multiply by a square root that will give you a perfect square under the radical when multiplied by the numerator.

If the radical in the numerator is a cube root, then you multiply by a cube root that will give you a perfect cube under the radical when multiplied by the numerator and so forth Example 3 : Rationalize the numerator.

Since we have a square root in the numerator, then we need to multiply by the square root of an expression that will give us a perfect square under the radical in the numerator.

AND Step 3: Simplify the fraction if needed. Also, we cannot take the square root of anything under the radical. Example 4 : Rationalize the numerator.

Since we have a cube root in the numerator, we need to multiply by the cube root of an expression that will give us a perfect cube under the radical in the numerator. Rationalizing the Denominator with two terms Above we talked about rationalizing the denominator with one term.

Again, rationalizing the denominator means to get rid of any radicals in the denominator. Because we now have two terms, we are going to have to approach it differently than when we had one term, but the goal is still the same.

Step 1: Find the conjugate of the denominator. You find the conjugate of a binomial by changing the sign that is between the two terms, but keep the same order of the terms. Step 2: Multiply the numerator and the denominator of the fraction by the conjugate found in Step 1. Step 3: Make sure all radicals are simplified. Step 4: Simplify the fraction if needed. Example 5 : Rationalize the denominator Step 1: Find the conjugate of the denominator.