Lesson Explainer: Rationalizing Denominators - Nagwa

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 1√2 can be rewritten to have a rational denominator by multiplying both its numerator and denominator by √2. We have 1√2=1×√2√2×√2=√22.

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 √2√2=1, once again not changing its value: 1√2=1√2×√2√2=√22.

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, 1√2 is the multiplicative inverse of √2, but written in the form √22, we see it is half the value of √2. This form also allows for easier addition and subtraction of these numbers. For example, if we want to evaluate 1√2+1√3, then we can do this by cross multiplication to get 1√2+1√3=√3+√2√2×√3=√3+√2√6.

However, if we rationalize the denominators first, we get an easier way to understand the result: 1√2+1√3=1√2×√2√2+1√3×√3√3=√22+√33=3√2+2√36.

We can follow this same process to rationalize the denominator of any fraction in the form 𝑎√𝑏, where 𝑎 is a real number and 𝑏 is a positive integer. We multiply the numerator and denominator by √𝑏 to get 𝑎√𝑏=𝑎×√𝑏√𝑏×√𝑏=𝑎√𝑏𝑏.

We have shown the following property.

Property: Rationalizing a Denominator

If 𝑎 is a real number and 𝑏 is a positive integer, then 𝑎√𝑏=𝑎×√𝑏√𝑏×√𝑏=𝑎√𝑏𝑏.

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 −1118√2 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 √2. We get −1118√2=−11×√218×√2×√2=−11√218×2=−11√236.

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 4√10−3√74√10 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 √10, we are going to want to multiply both the numerator and denominator by √10. We get 4√10−3√74√10=4√10−3√7×√104√10×√10.

In the denominator, we have 4√10×√10=40, and in the numerator, we can distribute √10 over the difference to get 4√10−3√7×√10=4√10×√10−3√7×√10=40−3√7×10=40−3√70.

Substituting these values in gives us 4√10−3√7×√104√10×√10=40−3√7040.

It is worth noting that 70=2×5×7, so it has no perfect square factors greater than 1. So, we cannot simplify this radical any further.

Hence, 4√10−3√74√10=40−3√7040.

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 1+√31+√2 and 21+√3. 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 𝑎 and 𝑏, we have 𝑎−𝑏=(𝑎−𝑏)(𝑎+𝑏).

We can use this result to rationalize both denominators. Substituting 𝑎=1 and 𝑏=√2 into this equation gives 1−√2=1−√21+√21−2=1−√21+√2−1=1−√21+√2.

Hence, we can rationalize the denominator of the first term by multiplying the numerator and denominator by 1−√2. This gives 1+√31+√2=1+√31+√2×1−√21−√2=1+√31−√21+√21−√2=1+√31−√2−1=−1+√31−√2.

Substituting 𝑎=1 and 𝑏=√3 into the difference of squares equation gives 1−√3=1−√31+√3−2=1−√31+√3.

Thus, 21+√3=21+√3×1−√31−√3=21−√31+√31−√3=21−√3−2=−1−√3.

We can now add these fractions together: 1+√31+√2+21+√3=−1+√31−√2−1−√3.

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 √𝑎+√𝑏, where 𝑎 and 𝑏 are positive rational numbers, we call √𝑎−√𝑏 its conjugate. Similarly, the conjugate of √𝑎−√𝑏 is √𝑎+√𝑏.

We extend this to the radical expression of the form 𝑎+𝑏√𝑐, where 𝑎, 𝑏, and 𝑐∈ℚ and 𝑐≥0 by saying its conjugate is 𝑎−𝑏√𝑐. 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, √𝑎 or √𝑏 may themselves be rational, and we can still apply the definition of a conjugate. For example, the conjugate of 5+√2 is 5−√2, and their product is still rational: 5+√25−√2=5−√2=23.

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 √𝑎+√𝑏 and its conjugate √𝑎−√𝑏: √𝑎+√𝑏+√𝑎−√𝑏=2√𝑎.

Another useful property is the product of a number and its conjugate: √𝑎+√𝑏√𝑎−√𝑏=√𝑎−√𝑏=𝑎−𝑏.

We can summarize these properties as follows.

Property: Properties of the Conjugate

For any radical expression √𝑎+√𝑏, where 𝑎 and 𝑏 are positive rational numbers,

• √𝑎+√𝑏+√𝑎−√𝑏=2√𝑎,
• √𝑎+√𝑏√𝑎−√𝑏=𝑎−𝑏.

Hence, we can rationalize denominators of the form √𝑎+√𝑏, where 𝑎 and 𝑏 are positive rational numbers, by multiplying the numerator and denominator by the conjugate: √𝑎−√𝑏. Similarly, we can rationalize denominators of the form √𝑎−√𝑏 by multiplying the numerator and denominator by √𝑎+√𝑏.

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 11+94√94? Express your answer in simplest form.

Answer

We first recall that the conjugate of the radical expression √𝑎+√𝑏, where 𝑎 and 𝑏 are positive rational numbers, is √𝑎−√𝑏. 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 √94 in the second term. We have 11+94√94=11+94√94×√94√94=11+94√9494=11+√94.

We note that 94=2×47 has no square prime factors, so √94 cannot be simplified further.

Now, we switch the sign of the radical term to find the conjugate. This gives us 11−√94.

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 −47√2−7=𝑎√2+𝑏, find the values of 𝑎 and 𝑏

Answer

To rewrite this expression in the form 𝑎√2+𝑏, 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 𝑎+𝑏√𝑐, where 𝑎, 𝑏, and 𝑐∈ℚ and 𝑐≥0, is 𝑎−𝑏√𝑐. So, we can start by rewriting √2−7 as −7+√2. Then, we can see that the conjugate of −7+√2 will be −7−√2. Therefore, we multiply the numerator and denominator by this conjugate to get −47√2−7×−7−√2−7−√2=−47−7−√2−7+√2−7−√2.

In the numerator, we can distribute −47 over the parentheses to get −47−7−√2=(7×47)+47√2=329+47√2.

In the denominator, we can either distribute or use the difference of squares to note that −7+√2−7−√2=7−√2=49−2=47.

Substituting these values into the expression gives −47−7−√2−7+√2−7−√2=329+47√247.

We then divide each term separately by 47, where we note that 32947=7. This gives us 329+47√247=7+√2.

Hence, 𝑎=1 and 𝑏=7.

In the previous example, we multiplied the numerator and denominator by the conjugate −7−√2. However, it is worth noting that the calculations can be made easier by instead multiplying the numerator and denominator by √2+7.

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 𝑥=1√3−√2 and 𝑦=√3−√2, find (𝑥−𝑦).

Answer

To evaluate this expression, we need to subtract √3−√2 from 1√3−√2, and the easiest way to do this is to first rationalize the denominator of the fraction 1√3−√2. 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 √3+√2. We can then rationalize the denominator as follows: 1√3−√2=1√3−√2×√3+√2√3+√2=√3+√2√3−√2√3+√2.

We can simplify the denominator of this expression by noting that √3−√2√3+√2=√3−√2=3−2=1.

Hence, √3+√2√3−√2√3+√2=√3+√21=√3+√2.

So, 𝑥=√3+√2 and 𝑦=√3−√2. Substituting these values into the expression gives (𝑥−𝑦)=√3+√2−√3−√2.

We can then distribute the factor of −1 and evaluate to get √3+√2−√3−√2=√3+√2−√3+√2=2√2=2×√2=4×2=8.

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 𝑥=√13−3 and 𝑦=√13+3, find the value of 𝑥𝑦+𝑦𝑥.

Answer

Multiplying by the reciprocal of a number is the same as dividing by this number, so we have 𝑥𝑦+𝑦𝑥=𝑥𝑦+𝑦𝑥.

Substituting the given values for 𝑥 and 𝑦, we get 𝑥𝑦+𝑦𝑥=√13−3√13+3+√13+3√13−3.

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 3−√13, and we will multiply the numerator and denominator of the second term by −3−√13 as follows: 𝑥𝑦+𝑦𝑥=√13−3√13+3+√13+3√13−3=√13−3√13+3×3−√133−√13+√13+3√13−3×−3−√13−3−√13=√13−33−√133+√133−√13+√13+3−3−√13−3+√13−3−√13.

We can evaluate each denominator as 3+√133−√13=3−√13=−4,−3+√13−3−√13=(−3)−√13=−4.

We now want to evaluate the numerators. In the first term, we have √13−33−√13=3√13−√13−3+3√13=3√13−13−9+3√13=−22+6√13.

In the second term, we have √13+3−3−√13=−3√13−√13−3−3√13=−3√13−13−9−3√13=−22−6√13.

Substituting in these all of these values and evaluating gives 𝑥𝑦+𝑦𝑥=√13−33−√133+√133−√13+√13+3−3−√13−3+√13−3−√13=−22+6√13−4+−22−6√13−4=−22+6√13+−22−6√13−4=−44−4=11.

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 𝑎 is a real number and 𝑏 is a positive integer, then 𝑎√𝑏=𝑎×√𝑏√𝑏×√𝑏=𝑎√𝑏𝑏.
• If 𝑎 and 𝑏 are positive rational numbers, then we say that √𝑎+√𝑏 and √𝑎−√𝑏 are conjugates.
• Similarly, if 𝑎, 𝑏, and 𝑐∈ℚ and 𝑐≥0, then the conjugate of 𝑎+𝑏√𝑐 is 𝑎−𝑏√𝑐. 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 √𝑎+√𝑏√𝑎−√𝑏=𝑎−𝑏.
• We can rationalize fractions with a denominator of √𝑎±√𝑏 by multiplying both the numerator and denominator of the fraction by the conjugate: √𝑎∓√𝑏.
• Rationalizing the denominators of fractions makes calculations involving fractions with square roots easier to evaluate.