In this explainer, we will learn how to rationalize square roots in the denominators of fractions.
There are many different forms we can use to represent surds, and these are useful for different calculations. For example, we can rewrite surds by taking perfect square factors out of the radicand. This process often makes the surds easier to add.
To see why this is the case, consider . In this form, it is difficult to add the surds together; however, we can note that and . Therefore, we can rewrite this sum as
This is much easier to simplify since both terms share a factor of :
Just as we can manipulate surds to make them easier to work with, there are techniques we can use to simplify surds when they appear in fractions. For example, consider . We can multiply both the numerator and denominator by to rewrite the expression to have a rational denominator:
We now have two different forms of the same number:
In the first form, it very easy to evaluate , since we can cancel the shared factors of to get
However, it is difficult to get an idea of the size of the number, since is tricky to evaluate mentally. In comparison, we can see that is half of the square root of two, which is much easier to approximate in our head.
Another advantage of the rationalized form is that it is more useful for addition. For example, consider . To add these fractions together, we want their denominators to be equal, so we need to cross multiply. This will then involve multiplying by both and , and we will get
This is once again a difficult expression to use in addition calculations and it is hard to wrap our head around the size of this number. Instead, we could add the fractions together if they were written with rational denominators. We can rewrite them in this form by multiplying both the numerator and denominator of each fraction by the surd in the denominator; this will have the same effect as multiplying by 1. We have
Similarly,
Adding these two fractions gives us
The result is a fraction with a rational denominator, so it is easier to combine with other fractions of this form. We call the process of rewriting fractions to have integer denominators “rationalizing the denominator.” In general, it is usually a good idea to rationalize the denominator before carrying out any operations on surds.
We can follow the same process to rationalize the denominator of any fraction in the form , where is a real number and is a positive integer. We multiply the numerator and denominator by to get
We have shown the following property.
Property: Rationalizing a Denominator
If is a real number and is a positive integer, then
This is known as rationalizing the denominator.
In our first example, we will use this process to rationalize the denominator of a given fraction.
Example 1: Simplifying a Fraction by Rationalizing the Denominator
Simplify by rationalizing the denominator.
Answer
Rationalizing the denominator means rewriting the value to have a rational number as its denominator. The denominator of the given value contains the surd ; thus, we can rewrite the value by multiplying by . This is equivalent to multiplying by 1, so it will not change its value. We have
We note that , so
We can rationalize the denominators of other fractions involving surds by using other results in algebra. For example, we know that we factor a difference of two squares as
If we substitute and into this equation, we get
In other words, we can multiply an expression of the form by to get an integer value.
We can use this result to rationalize any denominator of this form. For example, let’s rationalize the denominator of . We will multiply the numerator and denominator of the fraction by to get
We can simplify the numerator by expanding by using a difference of two squares:
Reversing the sign of the surd is called conjugation. So, we would say that is a conjugate of . In general, we can rationalize the denominators of this form by multiplying the numerator and denominator by the conjugate of the denominator. We define conjugates formally as follows.
Definition: Conjugate of a Radical Expression
We find the conjugate of a radical expression by switching the sign of the radical term, . If we have two radical terms, , then, we can switch the sign of either term; however, we usually switch the sign of the second term, .
In our next example, we will find a simplified form for the conjugate of a given radical expression.
Example 2: Finding the Conjugate of a Radical Expression
What is the conjugate of ? Express your answer in simplest form.
Answer
We first recall that the conjugate of a binomial means switching the sign of one of its terms. In a radical expression with two terms, we can find the conjugate by switching the sign of either term. Unless otherwise stated, we usually switch the sign of the surd, so a conjugate of is .
We are told to give our answer in its simplest form, so we need to rationalize the denominator. We can do this by multiplying both the numerator and denominator by . We have
In our next example, we will find the values of two unknowns in an equation by rationalizing the denominator of a radical expression.
Example 3: Rationalizing a Fraction to Find Unknown Values
Given that , find the values of and .
Answer
To find the values of and , we note that the right-hand side of the equation is a radical expression in a simplified form. We can determine the unknown values by rewriting the left-hand side of the equation in the same form. To do this, we need to first rationalize the denominator.
The denominator of is a radical expression, so we will multiply both the numerator and denominator of this fraction by the conjugate of the denominator. We can switch the sign of either the surd or the integer; however, in this case, it is easier to switch the sign of the integer. We have
In the denominator, we have the factored form of a difference of two squares, . So, we can expand the denominator to obtain
Substituting this into the expression yields
Thus,
We can see that the two sides of the equation are equal when and .
In our next example, we will simplify a fraction by rationalizing its denominator.
Example 4: Simplifying a Fraction by Rationalizing the Denominator
Simplify by rationalizing the denominator.
Answer
Rationalizing the denominator means rewriting the value to have a rational number as its denominator. In the given fraction, the denominator is a radical expression, so we will need to multiply both the numerator and denominator by the conjugate of this expression.
We find the conjugate by switching the sign of one of the surds. It is easier to switch the signs so that both terms are positive, so we will use . Multiplying both the numerator and denominator by this conjugate yields
In the denominator, we have the factored form of a difference of two squares, . Using this to expand the denominator, we obtain
Expanding the numerator and evaluating then yields
We can take out a factor of 4 in the numerator and cancel it out to find
Finally, we take out the shared factor of 9:
In our next example, we will simplify a radical expression by evaluating and rationalizing its denominator.
Example 5: Simplifying a Fraction by Rationalizing the Denominator
Simplify by rationalizing the denominator.
Answer
We cannot directly rationalize the denominator in the expression, since the denominator is not in a simplified form. Instead, we will start by evaluating the square; we need to expand the brackets:
We have
Thus,
We can now rationalize the denominator by multiplying both the numerator and denominator of the fraction by the conjugate of its denominator, which we find by switching the sign of the surd. This gives us
In the denominator, we have the factored form of a difference of two squares, . Using this to expand the denominator, we obtain
In our final example, we will simplify a radical expression by evaluating and rationalizing its denominator.
Example 6: Simplifying a Fraction by Rationalizing the Denominator
Simplify by rationalizing the denominator.
Answer
In order to rationalize the denominator of this expression, we first need to expand the brackets in the denominator. We have
Thus,
We can now rationalize the denominator by multiplying both the numerator and denominator of the fraction by the conjugate of its denominator, which we find by switching the sign of the surd. This gives us
In the denominator, we have the factored form of a difference of two squares, . Using this to expand the denominator, we obtain
We note that the numerator has a factor of , which it shares with the denominator:
Let’s finish by recapping some of the important points from this explainer.
Key Points
- We call the process of rewriting fractions to have integer denominators “rationalizing the denominator.”
- We can rationalize the denominators of fractions in the form by multiplying the numerator and denominator by . We have
- We can find a conjugate of a radical expression by switching the sign of the radical term, . If we have two radical terms: , then, we can switch the sign of either term; however, we usually switch the sign of the second term, .
- We can multiply a radical expression by its conjugate to remove the surd. In general,
- We can rationalize the denominators of fractions in the form by multiplying the numerator and denominator by the conjugate of the denominator, .
