Rationalizing the Denominator

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In this section, we learn how to rationalize the denominator. Recall that when we report a simplified radical, there can’t be a radical in the denominator. When a radical exists in any denominator, we use a process known as rationalizing the denominator. The easiest scenario occurs when there is a square root in the denominator. In this case, we can multiply the numerator and denominator by the square root. This will eliminate the square root from the denominator. We then can look to see what else can be done to simplify our answer. The more tedious scenario occurs with higher-level roots. When these occur, we must give some thought into what needs to be multiplied by our radical to achieve a rational number. Let’s take the example of a denominator which is the cube root of 4. In this case, we do not simply multiply the numerator and denominator by the cube root of 4. This would not leave a radical free denominator. Instead, we must think about 4 and see what is needed to obtain a perfect cube. 4 can be multiplied by 2, to obtain 8, which is a perfect cube (2 • 2 • 2). So for this scenario, we would multiply the numerator and denominator by the cube root of 2. This would give us a radical free denominator. Additionally, we will learn how to rationalize a binomial denominator (two terms) along with how to rationalize any denominator that is more than one single term. When we encounter a binomial denominator with a radical involved, we can’t simply multiply numerator and denominator by the radical. We begin by learning about the conjugate. The conjugate is obtained by keeping the terms the same and changing the sign between the two. Once we have obtained the conjugate, we can use this to rationalize a binomial denominator. We will multiply the numerator and denominator by the conjugate. This creates a scenario in the denominator that turns into the difference of two squares, once the multiplication is done. We will then move into a more advanced scenario, rationalizing a trinomial denominator. For us to rationalize a denominator with more than two terms, we rely on a grouping trick.
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