Rationalizing the Numerator

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  • 0:00 Rationalizing the Numerator
  • 1:36 Single Radical
  • 3:21 Conjugate Radical
  • 5:04 Lesson Summary
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Lesson Transcript
Instructor: Kimberly Osborn
Rationalizing the numerator of a fraction is necessary when you are working with an irrational number. This lesson will focus on identifying irrational numbers in your fraction and using that irrational number to manipulate your fraction.

Rationalizing the Numerator

When thinking about what you've learned about the different categories of numbers, you might remember the broad terms, rational and irrational numbers. It's important to understand how these two groups differ from each other before diving into rationalizing either the numerator or denominator of a fraction.

The major distinction between a rational and irrational number is whether or not a number can be written as the ratio of two integers, or whole numbers, in the form of decimals, fractions, or whole numbers. A rational number, no matter how large, meets this requirement, while an irrational number does not.

Irrational numbers can come in the form of a decimal with endless non-repeating digits or a non-perfect integral, such as an integral that does not spit out a whole number. A common example of an irrational number is pi, or 3.1415926….

In terms of integrals, the square root of four results in the whole number, two, making it a rational number. However, the square root of five results in the non-terminating and non-repeating decimal 2.23606 etc, making it an irrational number. Because we sometimes see fractions with an irrational number in the numerator or denominator, it is important to have a set of procedures to rationalize, or turn that irrational number into a rational one. While it is more common to rationalize a denominator, there are still cases where rationalizing the numerator is necessary.

Single Radical

Say you are given the following fraction and you are asked to rationalize your numerator.

Step #1: Take a look at your numerator and decide if it is irrational or rational. In this case, we have an irrational number, the square root of 8.

Step #2: Once you have determined that your numerator is irrational, you are going to create a new fraction using your irrational number as both the numerator and denominator. Remember, when you have the same number as the numerator and denominator, your fraction is equivalent to the number one.

Step #3: Because your created fraction is equivalent to one, and the identity property tells us that we can multiply any number by one and get the same number, you can multiply your original fraction by your created fraction and your answer will be equal to your original number.

Step #4: One of the rules of integrals tells us that when we multiply an integral by itself, the integral sign goes away, and we are left with the number inside of the integral. In the case of our numerator, we multiply the square root of 8 by the square root of 8 to eliminate our integral. Now we see why we need to multiply by our irrational number.

Step #5: This leaves us with a rational number in our numerator, but it is very important for us to NOT stop there. You must always simplify your answer! In the case of our solution, we can reduce the fraction by a factor of two to get our final answer with the numerator rationalized.

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