Stuck on how to rationalize the denominator? You’re not alone. Our comprehensive guide demystifies this concept, walking you through the process step-by-step. The review guide has many examples with square roots, cube roots, and multiple terms in the denominator. Let’s do this!
What We Review
What does it mean to rationalize the denominator?
To begin with, “rationalizing the denominator” is a common math practice that comes into play when we deal with fractions with a radical, such as a square root, in the denominator. When we talk about “rationalizing the denominator,” we’re referring to the process of adjusting the fraction so that the denominator becomes a rational number—a number that can be expressed as a simple fraction without any radicals.
Why do we need to rationalize denominators?
Remember: when we rationalize a denominator, we don’t actually change the fraction’s value. We rewrite the fraction in a way that’s easier to understand and use in math. Most algebra classes teach how to rationalize the denominator because it’s easier when the bottom of the fraction doesn’t have radicals (square roots, etc.).
Simply put: rationalizing the denominator makes fractions clearer and easier to work with. At first, it might seem slightly confusing, but it’s a helpful way to handle fractions in different math and even real-life situations.
Tip: This article reviews more detail the types of roots and radicals.
Now that you have context, let’s try some practice examples to learn how to rationalize the denominator!
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How to rationalize the denominator?
Here are the four steps that you can use to rationalize the denominator of a fraction:
Step 1: Identify the Radical in the Denominator
The first step is to identify if there is a radical in the denominator that needs to be rationalized. This could be a square root, cube root, or any other radical.
Step 2: Determine the Correct Rationalizing Factor
The rationalizing factor is what you’ll multiply the fraction by to eliminate the radical from the denominator.
For example, if the denominator is a single term with a square root, the rationalizing factor is usually the same as the denominator. If the denominator is a binomial (two terms) involving a square root, the rationalizing factor is the conjugate of the denominator. We’ll look at lots of examples below that show how this step works depending on the “type” of denominator we’re dealing with.
Step 3: Multiply the Fraction (by “1”) and Simplify
After that, we multiply the fraction’s numerator and denominator by the rationalizing factor. Remember, anything you do to the denominator of a fraction must also be done to the numerator to maintain the value of the fraction.
Must be remembered: since we’re multiplying the numerator and the denominator by the same value, we’re essentially just multiplying the fraction “by 1”. This is how we guarantee the original fraction and the “new” fraction are equivalent.
After multiplying, simplify the fraction if necessary. This could involve combining like terms, reducing the fraction to the lowest terms, or simplifying any radicals in the numerator.
And that’s it! While the specific steps might change slightly depending on the type of fraction you’re dealing with, this process provides a general guide on how to rationalize the denominator.
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How to rationalize the denominator with a square root? (examples)
We’ll start with the most basic examples of rationalizing denominators: working with square roots.
Example 1
Rationalize the denominator: \frac{3}{\sqrt{2}}.Step 1: Identify the radical in the denominator. Here, it’s \sqrt{2}.
Step 2: Determine the rationalizing factor. Since our denominator is just a single square root, the rationalizing factor is \sqrt{2}.
Tip: we choose this rationalizing factor because when we multiply \sqrt{2} by itself, the result is a rational number, 2, which eliminates the radical from the denominator.
Step 3: Multiply the fraction by the rationalizing factor. This means we multiply both the numerator and denominator by \sqrt{2}, like so:
\frac{3}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}
Simplifying gives us \frac{3\sqrt{2}}{2}.
Example 2
Now let's look at \frac{5}{\sqrt{5}}.Step 1: The radical in the denominator is \sqrt{5}.
Step 2: The rationalizing factor is also \sqrt{5}.
Remember: we choose this because when \sqrt{5} is multiplied by itself, we get 5, a rational number, which eliminates the radical in the denominator.
Step 3: Multiply the numerator and denominator by the rationalizing factor:
\frac{5}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}}
\frac{5\sqrt{5}}{5}
Simplifying our final answer: \sqrt{5}.
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Example 3
Finally, let's consider \frac{7}{\sqrt{3}}.Step 1: The radical in the denominator is \sqrt{3}.
Step 2: The rationalizing factor is \sqrt{3}. We select this because multiplying \sqrt{3} by itself gives us 3, a rational number, thereby removing the radical from the denominator.
Step 3: Multiply the numerator and denominator by the rationalizing factor:
\frac{7}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}}
\frac{7\sqrt{3}}{3}
For more examples, checkout the video below:
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How to rationalize the denominator with a cube root? (examples)
Rationalizing the denominator with a cube root follows a similar process to square roots. However, there’s an important difference.
Instead of multiplying by the same root to get a rational number, we need to raise the cube root to the power of 2. Why? The cube of any real number is rational, so by cubing the cube root, we eliminate the radical from the denominator. Let’s see this process in action with a couple of examples.
Example 1
Rationalize the denominator: \frac{4}{\sqrt[3]{2}}Step 1: Identify the radical in the denominator. Here, it’s \sqrt[3]{2}.
Step 2: Determine the rationalizing factor. In this case, we need the cube root to be raised to the power of 2 to get a rational number, so the rationalizing factor is \sqrt[3]{2^2}.
Step 3: Multiply the fraction by the rationalizing factor. This means we multiply both the numerator and denominator by \sqrt[3]{2^2}, like so:
\frac{4}{\sqrt[3]{2}} \times \frac{\sqrt[3]{2^2}}{\sqrt[3]{2^2}}
\frac{4\sqrt[3]{2^2}}{2}
Example 2
Next, let's consider \frac{5}{\sqrt[3]{3}}.Step 1: The radical in the denominator is \sqrt[3]{3}.
Step 2: The rationalizing factor is \sqrt[3]{3^2}. We choose this because when \sqrt[3]{3} is raised to the power of 3, we get 3, a rational number, which eliminates the radical in the denominator.
Step 3: Multiply the numerator and denominator by the rationalizing factor, like this:
\frac{5}{\sqrt[3]{3}} \times \frac{\sqrt[3]{3^2}}{\sqrt[3]{3^2}}
\frac{5\sqrt[3]{3^2}}{3}
Ready to push the difficulty even more? Keep reading!
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How to rationalize the square root of an entire fraction? (examples)
Before we jump into examples, let’s do a quick review of the Quotient Rule of Square Roots.
The Quotient Rule of Square Roots says the square root of a quotient equals the quotient of the square roots of the numerator and the denominator. In other words, this rule states that for any positive real numbers a and b:
\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}With that in mind, let’s jump into some examples:
Example 1
Consider the expression \sqrt{\frac{1}{2}}.Step 1: Identify the fraction under the radical and apply the Quotient Rule of Square Roots. In this case, the fraction under the radical is \frac{1}{2}, and applying the Quotient Rule of Square Roots, we get:
\sqrt{\frac{1}{2}} = \frac{\sqrt{1}}{\sqrt{2}}
Step 2: Determine the rationalizing factor. In this case, the denominator is \sqrt{2}, so our rationalizing factor is also \sqrt{2}.
Step 3: Multiply the fraction by rationalizing factor:
\frac{\sqrt{1}}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}}
\frac{\sqrt{2}}{2}
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Example 2
Now, let's look at simplifying \sqrt{\frac{3}{5}}.Step 1: The fraction under the radical is \frac{3}{5}. Applying the Quotient Rule of Square Roots gives us:
\sqrt{\frac{3}{5}} = \frac{\sqrt{3}}{\sqrt{5}}
Step 2: The rationalizing factor is \sqrt{5}, which is the square root in the denominator.
Step 3: Multiply the fraction to rationalize denominator:
\frac{\sqrt{3}}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}}
\frac{\sqrt{15}}{5}
In summary: by applying the Quotient Rule of Square Roots, we can simplify the problem and follow the same steps we’ve been using to rationalize the denominator. This rule is a key tool in our math toolbox that will help us simplify even more complex-looking problems!
For more examples, checkout this video:
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How to rationalize the denominator with two terms? (examples)
When we encounter a two-term denominator that contains a square root, we use a slightly different strategy for rationalization. This strategy involves using a mathematical concept called the “conjugate” of a binomial.
Conjugates of Binomials
In mathematics, the conjugate of a binomial is a two-term expression identical to the original, except the sign between the terms is reversed.
For example: the conjugate of a + \sqrt{b} is a - \sqrt{b}, and vice versa. The magic of conjugates lies in their multiplication: the product of a binomial and its conjugate is always a difference of squares, eliminating the square root from the denominator!
Let’s consider the binomial 3 + \sqrt{2}. Its conjugate is 3 - \sqrt{2}. When we multiply these two expressions, we get:
(3 + \sqrt{2})(3 - \sqrt{2}) = 9 - 2 = 7
Notice how the square root “disappeared” in the result? That’s the power of conjugates. In our examples below, we’ll use this concept to rationalize denominators with two terms.
Examples: Rationalizing Denominator with Two Terms
When rationalizing a denominator that contains two terms, we’ll follow the same four-step process as above, but with a twist in the second step. Instead of just using the square root in the denominator as our rationalizing factor, we’ll use the conjugate of the entire denominator. Let’s try it:
Example 1
Rationalize the denominator: \frac{2}{1 + \sqrt{3}}.Step 1: Identify the denominator to rationalize. Here, the denominator is 1 + \sqrt{3}.
Step 2: Determine the rationalizing factor. In this case, it’s the conjugate of the denominator, which is 1 - \sqrt{3}.
Step 3: Multiply the fraction by the rationalizing factor, remembering to multiply both the numerator and the denominator:
\frac{2}{1 + \sqrt{3}} \times \frac{1 - \sqrt{3}}{1 - \sqrt{3}}
First, we multiply the numerators:
2 \times (1 - \sqrt{3}) = 2 - 2\sqrt{3}
Next, we multiply the denominators, which are conjugates, to get a difference of squares:
(1 + \sqrt{3})(1 - \sqrt{3}) = 1 - 3 = -2
Putting it together, we get \frac{2 - 2\sqrt{3}}{-2}, which simplifies to:
\sqrt{3} - 1
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Example 2
Finally, let's look at \frac{5}{4 - \sqrt{2}}.Step 1: The denominator to rationalize is 4 - \sqrt{2}.
Step 2: The rationalizing factor is the conjugate of the denominator, which is 4 + \sqrt{2}.
Step 3: Now let’s multiply the fraction by our rationalizing factor:
\frac{5}{4 - \sqrt{2}} \times \frac{4 + \sqrt{2}}{4 + \sqrt{2}}
First, we multiply the numerators:
5 \times (4 + \sqrt{2}) = 20 + 5\sqrt{2}
Next, we multiply the denominators, which are conjugates, to get a difference of squares:
(4 - \sqrt{2})(4 + \sqrt{2}) = 16 - 2 = 14
Simplifying, we see:
\frac{20 + 5\sqrt{2}}{14}
\frac{10}{7} + \frac{5\sqrt{2}}{14}
Remember, the use of conjugates in rationalizing the denominator with two terms is a powerful tool. With practice, you’ll find it becomes second nature to you!
For a few more practice problems, checkout the video below:
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Conclusion
In conclusion, we’ve explored the process of rationalizing the denominator, a common technique in mathematics that can simplify complex fractions and make them easier to work with. Whether you’re dealing with a simple square root in the denominator, a cube root, the square root of an entire fraction, or a two-term denominator, the same core steps apply:
- Step 1: Identify the denominator that needs to be rationalized.
- Step 2: Determine the rationalizing factor. This could be the square root in the denominator, the cube root, or the conjugate of a two-term denominator.
- Step 3: Multiply the fraction by the rationalizing factor, remembering to apply this to both the numerator and the denominator. Simplify as much as possible.
More helpful math reviews:
- Simplifying Radicals and Radical Expressions: Review and Examples
- Full list of review guides from Albert.
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