 Fractions are frequently used in square root calculations. In the calculation of fractions, the denominator may contain the root symbol. In that case, if the root sign is still in the denominator, the answer is wrong.

When calculating the square root, if the denominator contains the radical symbol, it should be removed. This process is called rationalizing the denominator. In square root fractions, it is okay for the numerator to contain the square root, but it is subtle when the denominator contains the radical sign.

So how do we rationalize the denominator? And why do we need to rationalize the denominator?

By learning these, we will be able to understand the procedure for correctly rationalizing the denominator in mathematics. We will explain how to rationalize the denominator.

## If There Is Radical Symbols in the Denominator, Make Rationalizing

In a square root fraction, the denominator may contain radical sign. For example, this is the case in the following situation.

• $\displaystyle\frac{1}{\sqrt{2}}$

In this fraction, the denominator contains $\sqrt{2}$. So, let’s make the fraction with no root sign in the denominator. This is rationalizing the denominator Square roots are irrational numbers. In mathematics, it is not good to have irrational numbers in the denominator. So, we change the number in the denominator to a rational number instead of an irrational number. Moving the square root that exists in the denominator to the numerator. This is rationalizing the denominator.

In mathematics, if the root symbol remains in the denominator, the answer is incorrect. So be sure to rationalize the denominator.

### Procedure to Make the Square Root of the Denominator into an Integer

How do we rationalize the denominator? The way to do it is to multiply the denominator and the numerator by the same number. In fractions, it does not matter if we multiply or divide the denominator and the numerator by the same number. In fact, the following fractions are the same number.

• $\displaystyle\frac{1}{2}=\displaystyle\frac{2}{4}=\displaystyle\frac{3}{6}$

And we can think of it as $\displaystyle\frac{1}{2}=\displaystyle\frac{1×2}{2×2}$. Anyway, the fractions can be multiplied the denominator and the numerator by the same number.

Let’s take advantage of this property to rationalize the denominator. When squaring the root, the root symbol is removed. So we multiply the numerator and denominator by the same number as the square root number in the denominator. We get the following. $(\sqrt{2})^2=2$. We can convert the number of denominators to an integer by multiplying the same number of square roots. The way to rationalize the denominator is not difficult. All we have to do is multiply the square root in the denominator.

### Smaller Numbers in the Radical Symbol Is Less Likely to Make Miscalculation

When doing rationalizing the denominator, the smaller the number in the root symbol, the less likely it is to be miscalculated. We can reduce the number by doing prime factorization first.

For example, how can we rationalize the following fractions?

• $\displaystyle\frac{\sqrt{63}}{\sqrt{72}}$

When rationalizing the denominator, there are two ways to do it. One is to first reduce the number in the radical sign. The other way is to rationalize the denominator without first doing prime factorization. If we compare the two methods, we get the following. In this comparison, it is easier to calculate by first doing a prime factorization and then reducing the number in the root sign. In other words, there will be fewer calculation mistakes.

Any person can make mistakes in math calculations. So we have to choose ways to reduce the number of mistakes as much as possible. When we rationalize the denominator, if we first do prime factorization, we will make fewer calculation mistakes.

## Rationalizing of Addition and Subtraction with Two Terms in the Denominator

An advanced problem when doing rationalizing the denominator is the case where there are two terms in the denominator. There is not only one radical sign, but it contains addition or subtraction.

For example, the following fraction has two terms in its denominator.

• $\displaystyle\frac{1}{3-\sqrt{2}}$

In this case, how should we calculate it? If there are two terms, use the factoring formula. To be more specific, use the following formula.

• $(x+a)(x-a)=x^2-a^2$ By using this formula, we can rationalize the denominator. For example, the previous equation can be calculated as follows.

$\displaystyle\frac{1}{3-\sqrt{2}}$

$=\displaystyle\frac{1\textcolor{red}{×(3+\sqrt{2})}}{(3-\sqrt{2})\textcolor{red}{×(3+\sqrt{2})}}$

$=\displaystyle\frac{3+\sqrt{2}}{3^2-(\sqrt{2})^2}$

$=\displaystyle\frac{3+\sqrt{2}}{9-2}$

$=\displaystyle\frac{3+\sqrt{2}}{7}$

Thus, we can rationalize the denominator by using the factoring formula. Even if there are two terms, we can rationalize the denominator in the following way. In some cases, the denominator contains the addition or subtraction of the square root. In that case, use the factoring formula to rationalize the denominator.

## Why Do We Need to Rationalize the Denominator in Fractions?

Why does a square root fraction require rationalizing the denominator? What’s the problem with having a radical symbol in the denominator?

One reason why we need to rationalize the denominator is to make the numbers easier to understand. For example, it’s hard to understand what a number is in the following fraction.

• $\displaystyle\frac{1}{\sqrt{2}}$

Irrational numbers are numbers that last forever. It is impossible to divide the numerator by such a number and it is hard to imagine what kind of number it would be.

On the other hand, what if we do rationalizing the denominator and change the fraction to the following number.

• $\displaystyle\frac{\sqrt{2}}{2}$

Since it is an irrational number, the number in the numerator will last forever. However, we can easily imagine what kind of number it would be. $\sqrt{2}≒1.41$. So we can see that $\displaystyle\frac{\sqrt{2}}{2}$ is about 0.7.

If there are irrational numbers in the denominator, it is difficult to imagine the numbers. On the other hand, even if there is a square root, it is easier to understand what kind of number it is if it is in the numerator.

### When Making the Common Denominator, the Denominator needs to be Rationalized

Also, when making a common multiple of the denominators, it is preferable to rationalize the denominators. Addition and subtraction of fractions must have the same denominator. When the denominators are aligned, the properties of the fractions are the same.

However, if the denominator contains irrational numbers, it is difficult to align the denominators. For example, how can we do the following calculation?

• $\displaystyle\frac{1}{3}+\displaystyle\frac{1}{2+\sqrt{2}}$

In this state, the calculation is difficult. The numbers cannot be aligned because there is an irrational number in the denominator. So, let’s rationalize the denominator. Then we will be able to make the common denominator.

$\displaystyle\frac{1}{3}+\displaystyle\frac{1}{2+\sqrt{2}}$

$=\displaystyle\frac{1}{3}+\displaystyle\frac{1\textcolor{red}{×(2-\sqrt{2})}}{(2+\sqrt{2})\textcolor{red}{×(2-\sqrt{2})}}$

$=\displaystyle\frac{1}{3}+\displaystyle\frac{2-\sqrt{2}}{4-2}$

$=\displaystyle\frac{1}{3}+\displaystyle\frac{2-\sqrt{2}}{2}$

$=\displaystyle\frac{2}{6}+\displaystyle\frac{(2-\sqrt{2})×3}{6}$

$=\displaystyle\frac{2}{6}+\displaystyle\frac{6-3\sqrt{2}}{6}$

$=\displaystyle\frac{8-3\sqrt{2}}{6}$

If we make the denominator an integer, we can make common denominators of fractions. If an irrational number is in the denominator, it cannot make common denominators. Rationalizing the denominator is necessary because it is required to make common denominators so that the fractions can be calculated with each other. If we don’t rationalize the denominator, we can’t calculate it.

## Exercises: Calculation of rationalizing the denominator

Q1: Rationalize the denominator.

1. $\displaystyle\frac{4}{\sqrt{8}}$
2. $\displaystyle\frac{\sqrt{3}}{\sqrt{24}}$
3. $\displaystyle\frac{1}{\sqrt{6}+\sqrt{3}}$

If you understand what we have described so far, you can solve any problem easily.

(a)

$\displaystyle\frac{4}{\sqrt{8}}$

$=\displaystyle\frac{4}{2\sqrt{2}}$

$=\displaystyle\frac{2}{\sqrt{2}}$

$=\displaystyle\frac{2\textcolor{red}{×\sqrt{2}}}{\sqrt{2}\textcolor{red}{×\sqrt{2}}}$

$=\displaystyle\frac{2\sqrt{2}}{2}$

$=\sqrt{2}$

(b)

$\displaystyle\frac{\sqrt{3}}{\sqrt{24}}$

$=\displaystyle\frac{\sqrt{3}}{2\sqrt{6}}$

$=\displaystyle\frac{\sqrt{3}\textcolor{red}{×\sqrt{6}}}{2\sqrt{6}\textcolor{red}{×\sqrt{6}}}$

$=\displaystyle\frac{\sqrt{18}}{2×6}$

$=\displaystyle\frac{3\sqrt{2}}{12}$

$=\displaystyle\frac{\sqrt{2}}{4}$

(c)

$\displaystyle\frac{1}{\sqrt{6}+\sqrt{3}}$

$=\displaystyle\frac{1\textcolor{red}{×(\sqrt{6}-\sqrt{3})}}{(\sqrt{6}+\sqrt{3})\textcolor{red}{×(\sqrt{6}-\sqrt{3})}}$

$=\displaystyle\frac{\sqrt{6}-\sqrt{3}}{(\sqrt{6})^2-(\sqrt{3})^2}$

$=\displaystyle\frac{\sqrt{6}-\sqrt{3}}{3}$