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Lesson Explainer: Rationalizing Denominators of Radicals Mathematics

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 8+32. In this form, it is difficult to add the surds together; however, we can note that 8=4×2=22 and 32=16×2=42. Therefore, we can rewrite this sum as 8+32=22+42.

This is much easier to simplify since both terms share a factor of 2: 22+42=(2+4)2=62.

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 12. We can multiply both the numerator and denominator by 2 to rewrite the expression to have a rational denominator: 12=12×22=1×22×2=22.

We now have two different forms of the same number: 12=22.

In the first form, it very easy to evaluate 12×2, since we can cancel the shared factors of 2 to get 12×2=12×2=1.

However, it is difficult to get an idea of the size of the number, since 12 is tricky to evaluate mentally. In comparison, we can see that 22 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 12+13. 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 2 and 3, and we will get 12+13=12×33+13×22=36+26=3+26.

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 12=12×1=12×22=22.

Similarly, 13=13×1=13×33=33.

Adding these two fractions gives us 12+13=22+33=22×33+33×22=326+236=32+236.

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 𝑎𝑏, where 𝑎 is a real number and 𝑏 is a positive integer. We multiply the numerator and denominator by 𝑏 to get 𝑎𝑏=𝑎𝑏×𝑏𝑏=𝑎𝑏𝑏.

We have shown the following property.

Property: Rationalizing a Denominator

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

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 11182 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 2; thus, we can rewrite the value by multiplying by 22. This is equivalent to multiplying by 1, so it will not change its value. We have 11182=11182×22=11×2182×2.

We note that 2×2=2, so 11×2182×2=11218×2=11236.

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 (𝑝+𝑞)(𝑝𝑞)=𝑝𝑞.

If we substitute 𝑝=𝑎 and 𝑞=𝑏𝑐 into this equation, we get 𝑎+𝑏𝑐𝑎𝑏𝑐=𝑎𝑏𝑐=𝑎𝑏𝑐.

In other words, we can multiply an expression of the form 𝑎+𝑏𝑐 by 𝑎𝑏𝑐 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 11+2. We will multiply the numerator and denominator of the fraction by 12 to get 11+2=11+2×1=11+2×1212=121+212.

We can simplify the numerator by expanding by using a difference of two squares: 121+212=1212=1212=121=1+2.

Reversing the sign of the surd is called conjugation. So, we would say that 𝑎𝑏𝑐 is a conjugate of 𝑎+𝑏𝑐. 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 𝑎+𝑏𝑐 by switching the sign of the radical term, 𝑎𝑏𝑐. If we have two radical terms, 𝑎+𝑏, then, we can switch the sign of either term; however, we usually switch the sign of the second term, 𝑎𝑏.

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 11+9494? 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 11+9494 is 119494.

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 94. We have 119494=119494×9494=11949494=11949494=1194.

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 4727=𝑎2+𝑏, find the values of 𝑎 and 𝑏.

Answer

To find the values of 𝑎 and 𝑏, 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 4727 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 4727=4727×2+72+7=472+7272+7.

In the denominator, we have the factored form of a difference of two squares, (𝑎𝑏)(𝑎+𝑏)=𝑎𝑏. So, we can expand the denominator to obtain 272+7=27=249=47.

Substituting this into the expression yields 472+7272+7=472+747=2+7.

Thus, 2+7=𝑎2+𝑏.

We can see that the two sides of the equation are equal when 𝑎=1 and 𝑏=7.

In our next example, we will simplify a fraction by rationalizing its denominator.

Example 4: Simplifying a Fraction by Rationalizing the Denominator

Simplify 3626 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 2+6. Multiplying both the numerator and denominator by this conjugate yields 3626=3626×2+62+6=362+6262+6.

In the denominator, we have the factored form of a difference of two squares, (𝑎𝑏)(𝑎+𝑏)=𝑎𝑏. Using this to expand the denominator, we obtain 362+6262+6=362+626.

Expanding the numerator and evaluating then yields 362+626=362+36626=362+3664.

We can take out a factor of 4 in the numerator and cancel it out to find 362+3624=492+964=92+96.

Finally, we take out the shared factor of 9: 92+96=92+6.

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 123 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: 123=12323.

We have

Thus, 123=1743.

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 1743=1743×7+437+43=7+437437+43.

In the denominator, we have the factored form of a difference of two squares, (𝑎𝑏)(𝑎+𝑏)=𝑎𝑏. Using this to expand the denominator, we obtain 7+437437+43=7+43743=7+434948=7+43.

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 1173+7 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 173+7=(1×3)+1×7+7×3+7×7=337+77=3277=427.

Thus, 1173+7=1427.

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 1427=1427×4+274+27=4+274274+27.

In the denominator, we have the factored form of a difference of two squares, (𝑎𝑏)(𝑎+𝑏)=𝑎𝑏. Using this to expand the denominator, we obtain 4+274274+27=4+27(4)27=4+271627=4+271628=4+2712.

We note that the numerator has a factor of 2, which it shares with the denominator: 4+2712=22712=276.

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 𝑎𝑏 by multiplying the numerator and denominator by 𝑏. We have 𝑎𝑏=𝑎𝑏×𝑏𝑏=𝑎𝑏𝑏.
  • We can find a conjugate of a radical expression 𝑎+𝑏𝑐 by switching the sign of the radical term, 𝑎𝑏𝑐. If we have two radical terms: 𝑎+𝑏, then, we can switch the sign of either term; however, we usually switch the sign of the second term, 𝑎𝑏.
  • We can multiply a radical expression by its conjugate to remove the surd. In general, 𝑎+𝑏𝑐𝑎𝑏𝑐=𝑎𝑏𝑐.
  • We can rationalize the denominators of fractions in the form 𝑑𝑎+𝑏𝑐 by multiplying the numerator and denominator by the conjugate of the denominator, 𝑎𝑏𝑐.

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