In this explainer, we will learn how to rationalize square roots in the denominators of fractions.
In general, rationalizing the denominator means rewriting a fraction to have a rational number as its denominator. For example, the real number can be rewritten to have a rational denominator by multiplying both its numerator and denominator by . We have
It is worth noting that multiplying the numerator and denominator of a fraction by the same number will not change its value. We can also think of this as multiplying a fraction by , once again not changing its value:
We often think of this form of number as a simplified form of the radical numbers since it is easier to comprehend the value of these numbers. For example, is the multiplicative inverse of , but written in the form , we see it is half the value of . This form also allows for easier addition and subtraction of these numbers. For example, if we want to evaluate , then we can do this by cross multiplication to get
However, if we rationalize the denominators first, we get an easier way to understand the result:
We can follow this 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.
Let’s now see some examples of applying this process to rationalize the denominators of given fractions.
Example 1: Simplifying a Fraction by Rationalizing the Denominator
Simplify by rationalizing the denominator.
Answer
We first recall that rationalizing the denominator means we need to rewrite this fraction with a rational denominator. We can do this by multiplying both the numerator and denominator by . We get
Since 11 and 36 share no common factors other than 1, we cannot simplify any further.
In our next example, we will rationalize the denominator of a fraction with multiple terms in the numerator.
Example 2: Simplifying a Fraction by Rationalizing the Denominator
Simplify by rationalizing the denominator.
Answer
We begin by recalling that rationalizing the denominator means we need to rewrite this fraction with a rational denominator. Since the denominator is a single term containing , we are going to want to multiply both the numerator and denominator by . We get
In the denominator, we have , and in the numerator, we can distribute over the difference to get
Substituting these values in gives us
It is worth noting that , so it has no perfect square factors greater than 1. So, we cannot simplify this radical any further.
Hence,
Another reason we rationalize the denominators of fractions is that the process of cross multiplying to equate the denominators is easier if both denominators are rational. This process becomes even clearer if the denominators are even more complicated than just square roots.
For example, it is hard to find a common denominator between and . This means it is much more difficult to add these fractions together. Instead, we should rewrite these fractions to have rational denominators since this will make the cross multiplication process easier.
We can rationalize denominators in this form by recalling the difference of squares method of factoring. This tells us that for any real numbers and , we have
We can use this result to rationalize both denominators. Substituting and into this equation gives
Hence, we can rationalize the denominator of the first term by multiplying the numerator and denominator by . This gives
Substituting and into the difference of squares equation gives
Thus,
We can now add these fractions together:
We could simplify this further by distributing the parentheses.
To rationalize the denominators in the above example, we multiplied by each denominator where we switched the sign of the radical; this is called its conjugate. We define the conjugate as follows.
Definition: Conjugate of a Radical Expression
For the radical expression , where and are positive rational numbers, we call its conjugate. Similarly, the conjugate of is .
We extend this to the radical expression of the form , where , , and and by saying its conjugate is . In general, the conjugate is found by switching the sign of the radical part of the expression, and if the expression consists of two radical terms, we switch the sign of the second term.
We also note that, in the above definition, or may themselves be rational, and we can still apply the definition of a conjugate. For example, the conjugate of is , and their product is still rational:
The definition of a conjugate allows us to consider some of the properties of a number and its conjugate. For example, consider the sum of and its conjugate :
Another useful property is the product of a number and its conjugate:
We can summarize these properties as follows.
Property: Properties of the Conjugate
For any radical expression , where and are positive rational numbers,
- ,
- .
Hence, we can rationalize denominators of the form , where and are positive rational numbers, by multiplying the numerator and denominator by the conjugate: . Similarly, we can rationalize denominators of the form by multiplying the numerator and denominator by .
Let’s see an example of determining the conjugate of a given radical expression.
Example 3: 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 the radical expression , where and are positive rational numbers, is . Since the expression we are given is not in this form, we should start by rationalizing the denominator. We can do this by multiplying the numerator and denominator by in the second term. We have
We note that has no square prime factors, so cannot be simplified further.
Now, we switch the sign of the radical term to find the conjugate. This gives us .
In our next example, we will rewrite a given rational expression into a simplified form to find the values of unknown coefficients.
Example 4: Rationalizing a Fraction to Find Unknown Values
Given that , find the values of and
Answer
To rewrite this expression in the form , we are going to need to rationalize the denominator. We can do this by multiplying both the numerator and denominator of the fraction by the conjugate of the denominator.
We recall that the conjugate of the radical expression , where , , and and , is . So, we can start by rewriting as . Then, we can see that the conjugate of will be . Therefore, we multiply the numerator and denominator by this conjugate to get
In the numerator, we can distribute over the parentheses to get
In the denominator, we can either distribute or use the difference of squares to note that
Substituting these values into the expression gives
We then divide each term separately by 47, where we note that . This gives us
Hence, and .
In the previous example, we multiplied the numerator and denominator by the conjugate . However, it is worth noting that the calculations can be made easier by instead multiplying the numerator and denominator by .
In our next example, we will see how rationalizing a denominator can make calculations involving the quotients of radicals easier to evaluate.
Example 5: Finding the Value of an Algebraic Expression Involving Square Roots
Given that and , find .
Answer
To evaluate this expression, we need to subtract from , and the easiest way to do this is to first rationalize the denominator of the fraction . We can do this by multiplying both the numerator and denominator by the conjugate of the denominator. We change the subtraction in the denominator to an addition, which gives us . We can then rationalize the denominator as follows:
We can simplify the denominator of this expression by noting that
Hence,
So, and . Substituting these values into the expression gives
We can then distribute the factor of and evaluate to get
In our final example, we will evaluate an expression by rationalizing two denominators.
Example 6: Finding the Value of an Algebraic Expression Involving Square Roots and Division
Given that and , find the value of .
Answer
Multiplying by the reciprocal of a number is the same as dividing by this number, so we have
Substituting the given values for and , we get
We can evaluate this addition by rationalizing both denominators. To do this, we need to multiply the numerator and denominator of each term by the conjugate of their respective denominators. We find the conjugate of each denominator by switching the sign of the radical term.
So, we will multiply the numerator and denominator of the first term by , and we will multiply the numerator and denominator of the second term by as follows:
We can evaluate each denominator as
We now want to evaluate the numerators. In the first term, we have
In the second term, we have
Substituting in these all of these values and evaluating gives
Let’s finish by recapping some of the important points from this explainer.
Key Points
- Rationalizing the denominator means rewriting a fraction to have a rational number as its denominator. In general, if is a real number and is a positive integer, then
- If and are positive rational numbers, then we say that and are conjugates.
- Similarly, if , , and and , then the conjugate of is . In general, the conjugate is found by switching the sign of the radical part of the expression, and if the expression consists of two radical terms, we switch the sign of the second term.
- We note that the product of a radical expression with its conjugate is rational since
- We can rationalize fractions with a denominator of by multiplying both the numerator and denominator of the fraction by the conjugate: .
- Rationalizing the denominators of fractions makes calculations involving fractions with square roots easier to evaluate.
