Rationalizing Denominators: Study with Video Lessons, Practice Problems & Examples

To rationalize a denominator with a single radical, multiply both the numerator and denominator by the radical. For example, to simplify \( \frac{1}{\sqrt{3}} \), multiply by \( \sqrt{3} \) to get \( \frac{\sqrt{3}}{3} \). For a two-term denominator like \( 2 + \sqrt{3} \), use the conjugate \( 2 - \sqrt{3} \) to eliminate the radical, resulting in \( \frac{2 - \sqrt{3}}{1} \). This process ensures the denominator is a rational number, adhering to mathematical conventions.

Introduction to Trigonometry

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Rationalizing Denominators

Hey, everyone. Welcome back. So we've multiplied and divided radicals, and one of the things that we should know about radicals is that they can never be left in the bottom of a fraction. This is one of those weird, bad things that you just can't do in math. Now, you might be thinking we've already seen radicals in the bottoms of fractions like √2 of √8. But in that case, it was fine because usually those fractions reduced to perfect squares like \( \frac{1}{4} \). And then if it was a perfect square, the radical just goes away, and you're left with a rational number. What I'm going to show you in this video is that sometimes that doesn't happen. Sometimes you might have an expression like \( \frac{1}{\sqrt{3}} \), and you can't simplify that to a perfect square. So to solve these types of problems, we're going to have to do another thing. We're going to have to do something called rationalizing the denominator. I'm going to show you what that process is. It's actually really straightforward, so let's just go ahead and get to it. Again, if we have something like √2 of √8, it's simplified to a perfect square, and that was perfectly fine. So radicals can simplify to perfect squares, and we don't have to do anything else because you're just left with something like \( \frac{1}{2} \).

But, if you can't simplify this radical over here to a perfect square, then we're going to have to make it 1. And the way we make it 1 is by doing this thing called rationalizing the denominator. It's actually really straightforward. Basically, we're going to take this expression over here, and we're going to multiply it by something to get rid of that radical on the bottom. So, what you're going to do is you're going to multiply the top and the bottom, the numerator and the denominator, by something, and usually, that something that you multiply by is just whatever is on the bottom radical. So, in other words, we're going to take this expression over here, and I'm just going to multiply it by √3, but I have to do it on the top and the bottom. You always have to make sure to do it on the top and the bottom because then you're basically just multiplying this expression by 1, and you're not changing the value of it. So whatever you do at the bottom, you have to do on the top. And the reason this works is because let's just work it out. What is √3 times √3? Basically, once we've done this, we've now turned the bottom into a perfect square. It's the square root of 9, which we know is actually just 3. So in other words, we've multiplied it by itself to sort of get rid of the radical, and now it's just a rational number on the bottom.

Alright? So what happens to the top? Well, again, we just multiply straight across, and then we ended up with √3 over √9, which is just √3 over 3. So, look at the difference between where we started and ended. Here, we had \( \frac{1}{\sqrt{3}} \), we had a radical on the bottom. And here, when we're done, we actually have 3 on the bottom, and that's perfectly fine. We have a radical on top, but we can have radicals on the top, and that's perfectly fine. So what I want you to do is I actually want you to plug in, if you have a calculator handy, 1 divided by √3. When you plug this in, what you should get out of the calculator is 0.57. And now if you actually do √3 over 3, you're going to get the exact same numbers, 0.57. So the whole thing here is that these two expressions are exactly equivalent. They mean the exact same thing. It's just that in one case, we've gotten rid of the radical on the bottom. So this is what rationalizing the denominator means. Thanks for watching, and let's move on to the next one.

Rationalize the denominator. 
$-\frac{5}{2\sqrt7}$

• A. $-\frac{5}{14}$
• B. $-\frac{5\sqrt7}{2}$
• C. $-\frac{5\sqrt7}{14}$
• D. $-\frac{10\sqrt7}{14}$

Answer: (C)

Rationalize the denominator.
$\frac{6+\sqrt{x}}{-\sqrt{x}}$

• A. $\frac{6\sqrt{x}}{-x}-1$
• B. $6\sqrt{x}+x$
• C. $\frac{6\sqrt{x}}{x}+1$
• D. $\frac{7\sqrt{x}}{-x}$

Answer: (A)

Rationalizing Denominators Using Conjugates

Welcome back, everyone. We saw how to rationalize the denominator by taking something like 1 over radical 3. We multiply the top and the bottom by whatever was on the bottom, and we end up undoing the radical, and that was awesome. We could do that. So, what I'm going to show you in this video is that sometimes that won't happen. You might have a problem that looks like 1 over 2 + radical 3, in which you actually have 2 terms in the denominator, and when the denominator has 2 terms, multiplying by the same radical won't actually eliminate it. So what I'm going to show you is, actually, we're going to need something else to rationalize the denominator. We're going to need something called the conjugates. Rather than tell you, let me just go ahead and show you. Let's just jump into our problem here.

So why can't we just use 2 +3 on the bottom? Well, if you end up doing 2 +3, then when you multiply across, this actually ends up being a binomial multiplied by a binomial. So in other words, we actually have to foil. And if you foil this out, what's going to happen is, on the bottom, you're going to get 2 times 2. That's the first. We're going to get 2 times radical 3. Those are the outer terms. Same thing for the inner terms. So, in other words, you end up with 4 radical 3. And on the inner terms, radical 3 times itself will just be 3. So, in other words, I multiplied it by itself on the top and the bottom, but I still ended up with a radical on the bottom. I didn't get rid of that, and, remember, that's bad. You can't have radicals on the bottom. So multiplying it by itself is not going to work here. So what do we do? Well, instead of multiplying by itself, we do something. We multiply by what's called the conjugate of the bottom. Basically, what the conjugate is is you're just going to reverse the sign between the two terms. So, for example, if I have 2 +3, then the conjugate is just going to be 2 minus radical 3. That's the conjugate. You just take the sign between the terms and you flip it. So the general formula is if you have something like a + radical b, then the conjugate is going to be a minus radical b and vice versa. Those two things are conjugates of each other.

So what do we do here? Well, I'm just going to rewrite this. 1 over 2 +3. Now we're going to multiply not by itself. We're going to multiply by the conjugate, so in other words, 2 minus radical 3. And remember, whatever we multiply on the bottom, we have to multiply on the top. It has to be the same thing. So why does this work? Well, if you notice here, we're actually multiplying the same exact terms, 2 and radical 3, except the signs are flipped between them, and this actually ends up being a difference of squares. So remember how a difference of squares works? Basically, what happens is that you square these two numbers, so 2² becomes 4, and you square these two numbers, and then you basically just stick a minus sign between them. So, in other words, radical 3, when you square, it just becomes 3, and this is just a difference of squares. And on the top, when you multiply straight across, this ends up being 2 minus radical 3. Alright? So what do you end up with? You end up with 2 minus radical 3 divided by just 1. Now you won't always get 1 here. We just got 1 because we had 4 - 3. But, basically, what happens is that we just got rid of the radical. So, in other words, we've rationalized the denominator here. So multiplying by multiplying a radical by its conjugate, in fact, always eliminates the radical. That's why it's super useful, and it always just results in a rational number, like 1 or something like that. So just let me summarize really quickly here. When you have a one-term denominator, you multiply the top and the bottom by whatever is on the bottom. And when you have a 2-term denominator, you multiply by the conjugate of the bottom. Alright? But these are just the 2 ways that you rationalize the denominator. Hopefully, that made sense. Let me know if you have any questions. Thanks for watching.

Rationalize the denominator and simplify the radical expression.
$\frac{\sqrt7}{5-\sqrt6}$

• A. $\frac{\sqrt7}{19}$
• B. $\frac{5\sqrt7+\sqrt{42}}{-11}$
• C. $\frac{5\sqrt7+\sqrt{42}}{21}$
• D. $\frac{5\sqrt7+\sqrt{42}}{19}$

Answer: (D)

Rationalize the denominator and simplify the radical expression.
$\frac{2-\sqrt3}{2+\sqrt3}$

• A. $7-4\sqrt3$
• B. $\frac{1}{7+4\sqrt3}$
• C. $7+4\sqrt3$
• D. $\frac{1}{7-4\sqrt3}$

Answer: (A)

Here’s what students ask on this topic:

How do you rationalize the denominator of a fraction with a single radical?

To rationalize the denominator of a fraction with a single radical, you multiply both the numerator and the denominator by the radical. For example, to simplify $\frac{1}{\sqrt{3}}$, you multiply by $\sqrt{3}$ to get $\frac{\sqrt{3}}{3}$. This process eliminates the radical in the denominator, making it a rational number. The key is to multiply by a form of 1 that will clear the radical from the denominator.

What is the conjugate of a binomial with a radical, and how is it used in rationalizing the denominator?

The conjugate of a binomial with a radical is formed by changing the sign between the two terms. For example, the conjugate of $2+\sqrt{3}$ is $2-\sqrt{3}$. To rationalize the denominator of a fraction like $\frac{1}{2+\sqrt{3}}$, you multiply both the numerator and the denominator by the conjugate, resulting in $\frac{2-\sqrt{3}}{1}$. This process eliminates the radical in the denominator.

Why is it important to rationalize the denominator in mathematical expressions?

Rationalizing the denominator is important because it simplifies the expression and adheres to mathematical conventions. Having a rational number in the denominator makes the expression easier to understand and work with, especially in further calculations. It also ensures consistency in mathematical notation and avoids the complications that can arise from having radicals in the denominator.

Can you provide an example of rationalizing a denominator with a binomial radical?

Sure! Let's rationalize the denominator of $\frac{1}{2+\sqrt{5}}$. The conjugate of $2+\sqrt{5}$ is $2-\sqrt{5}$. Multiply both the numerator and the denominator by this conjugate:

$\frac{1\times 2-\sqrt{5}}{2+\sqrt{5}\times 2-\sqrt{5}}$

This results in:

$\frac{2-\sqrt{5}}{4-5}$

Simplifying the denominator, we get:

$\frac{2-\sqrt{5}}{-1}$

So, the rationalized form is $\frac{2-\sqrt{5}}{-1}$.

What happens if you don't rationalize the denominator in a fraction?

If you don't rationalize the denominator in a fraction, the expression may be more difficult to work with in further calculations. It can also lead to inconsistencies in mathematical notation and make it harder to compare and simplify expressions. Rationalizing the denominator ensures that the expression is in its simplest and most standard form, making it easier to understand and use in subsequent mathematical operations.