Rationalizing the Denominator: A Guide for Students

Why Rationalize?

I guess the easy answer would be to say that that’s just how it’s done. We rationalize the denominators because that is how it has always been done. But that’s not really a satisfying answer to the question. The actual reason goes back many years.

But before we get into the process of rationalizing the denominator, it’s crucial to understand what makes a denominator irrational. A denominator is said to be irrational when it contains an irrational number. An irrational number is a number that cannot be expressed as a simple fraction, meaning it doesn’t have a finite decimal representation and that non-finite decimal representation doesn’t ever repeat.

This usually occurs when the denominator contains a square root (or any root) of a number that isn’t a perfect square (or perfect cube, fourth power, and so on). For example, in the fraction frac{1}{sqrt{2}}​, the denominator sqrt{2} ​ is an irrational number because it doesn’t result in a neat, finite decimal but rather an unending, non-repeating decimal; 1.414213… .  Therefore, the denominator of this fraction, frac{1}{sqrt{2}}​, needs to be rationalized — but why?

From a historical perspective, the preference for a rational denominator dates back to the time when most computations were performed by hand. Having a rational denominator (as opposed to an irrational one) made arithmetic simpler and less prone to errors. Even with the advent of calculators and computers, this preference has persisted due to the aesthetic and clarity it lends to mathematical notation. We also continue to prefer fractions written with rational denominators because it allows for easier comparisons between fractions.

While it may seem that with today’s technology we don’t need to adhere strictly to this practice, it still holds importance in many areas of mathematics where certain manipulations require rational denominators.  It’s a good skill to have in your mathematical arsenal!

The Process of Rationalizing the Denominator

Now that we understand what it means to rationalize the denominator and why it’s important, let’s take a look at the process of rationalizing the denominator.

Case Study 1: Rationalizing a single term denominator: 

Let’s say we have a fraction like {large frac{1}{sqrt{2}} }. We can rationalize the denominator by multiplying both the numerator and the denominator by {large sqrt{2} }. This gives us {large frac{sqrt{2}}{2} }, a fraction with a rational denominator.

{large begin{aligned} frac{1}{sqrt{2}} cdot frac{sqrt{2}}{sqrt{2}} &=frac{1cdot sqrt{2}}{sqrt{2} cdot sqrt{2}} \ &= frac{sqrt{2}}{2} end{aligned} }

Another way to understand this process is to write the radical in the denominator as a fractional exponent.

{large begin{aligned} text{*Note; }x^{frac{a}{b}} &= sqrt[b]{x^{a}}  \ &= (sqrt[b]{x})^{a} \ \ thereforesqrt{2}&=2^{frac{1}{2}} \ text{and; }frac{1}{sqrt{2}} &=frac{1}{2^{frac{1}{2}}} \ \ frac{1}{2^{frac{1}{2}}} cdot frac{2^{frac{1}{2}}}{2^{frac{1}{2}}} &= frac{2^{frac{1}{2}}}{2^{frac{1}{2}+frac{1}{2}}} \ &= frac{sqrt{2}}{2} end{aligned} }

Case Study 2: Rationalizing a binomial denominator:

We are going to be taking advantage of the pattern of the difference of squares;

{large a^{2}-b^{2}=(a-b)(a+b)}

That is, the difference of squares pattern shows us that the product of binomial conjugates equals a difference of two squares. This is useful because we know that if we square a square root, we get a whole number.

{large sqrt{x} cdot sqrt{x}=x text{ for } xge0 }

Examle:  To rationalize the denominator in this fraction {Large frac{1}{1-sqrt{2}} }, we would multiply the numerator and denominator by {large (1+sqrt{2}) }

{large begin{aligned} frac{1}{1-sqrt{2}} cdot frac{1+sqrt{2}}{1+sqrt{2}} &= frac{1+sqrt{2}}{(1+sqrt{2})(1+sqrt{2})} \ &= frac{1+sqrt{2}}{-1} \ &= -sqrt{2} end{aligned} }

We are essentially multiplying the original fraction by a value of 1 which does not change the value of the fraction -– we are only changing the way that it looks.

This will always work if we have a denominator of a binomial in which one or both of the terms are square roots.

Common Pitfalls and Mistakes to Avoid

When rationalizing the denominator, there are a few common mistakes to avoid. One error is forgetting to multiply both the numerator and the denominator by the same term/expression, which changes the value of the fraction. Another frequent mistake is not properly simplifying after rationalizing. Keeping an eye out for these common errors can help improve your accuracy and efficiency in rationalizing denominators.

Practice Makes Perfect

To further cement your understanding, here are a few practice problems. Attempt to rationalize the denominators on your own. Then come back and check your work. Did you come to the same solution?:

1. Rationalize the denominator of {large frac{5}{sqrt{3}} }.

{large begin{aligned} text{Solution; } frac{5}{sqrt{3}} cdot frac{sqrt{3}}{sqrt{3}}&=frac{5cdot sqrt{3}}{sqrt{3} cdot sqrt{3}} \ &= frac{5sqrt{2}}{3} end{aligned} }

2. Rationalize the denominator of {large frac{-12}{sqrt{5}} }.

{large begin{aligned} text{Solution; } frac{-12}{sqrt{5}} cdot frac{sqrt{5}}{sqrt{5}}&=frac{-12cdot sqrt{5}}{sqrt{5} cdot sqrt{5}} \ &= frac{-12sqrt{5}}{5} end{aligned} }

3. Rationalize the denominator of {large frac{2}{1-sqrt{7}} }.

{large begin{aligned} text{Solution; } frac{2}{1-sqrt{7}} cdot frac{1+sqrt{7}}{1+sqrt{7}} &= frac{2(1+sqrt{7})}{(1-sqrt{7})(1+sqrt{7})} \ &= frac{2+2sqrt{7}}{1^{2}-(sqrt{7})^{2}} \ &= frac{2+2sqrt{7}}{1-7} \ &= frac{2+2sqrt{7}}{-6} \ &= frac{1+sqrt{7}}{-3} text{ or } frac{-1-sqrt{7}}{3}end{aligned} }

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Can a Math Tutor Help?

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