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# Intro to rationalizing the denominator

When we have a fraction with a root in the denominator, like 1/√2, it's often desirable to manipulate it so the denominator doesn't have roots. To do that, we can multiply both the numerator and the denominator by the same root, that will get rid of the root in the denominator. For example, we can multiply 1/√2 by √2/√2 to get √2/2. Created by Sal Khan and CK-12 Foundation.

## Want to join the conversation?

• Is it possible to rationalize a/pi?
• So the question is really, "why can we use this method with some irrational numbers like 1/√2 and 1/√111 but not with other irrational numbers like 1/pi?"

The difference is that you can use this technique for numbers under a radical that can be multiplied by other numbers under a radical to produce a whole number. √2 is irrational because there is no nice clean number multiplied by itself to become 2--the square root of 2 goes on forever like pi.
√2 = 1.41421356237...........
Same with many other numbers, both primes and composites. √5 and √38 are two more examples, as is the cube root of 25, ³√25. But if you multiply the square root of 2 times the square root of 2, you just get 2. And THAT is when we can use this method.

So there is a limit to the cases where you can use it. If you try to multiply pi by itself, this magic doesn't happen--you still have a mess of decimal places and an irrational result in the denominator.
• he gets this answer -24-12√5 for the third question. how are there 2 negative signs? shouldn't there be only 1 negative sign from the -1?
• - difference of squares?
• How to rationalize cube root denominators?
• Great question! Cube Root/nth root denominators can be rationalized using a very similar method to square root denominators. All you need to do is multiply both the top and bottom of the fraction by the Cube Root/nth root of the radicand (stuff inside of the radical) to the power of the index (3 for cube root denominators).

For example, we can simplify 1/cubeRoot(2), by multiplying both the top and the bottom by cubeRoot(2^2) which is equal to cubeRoot(4)/cubeRoot(8)=cubeRoot(4)/2.

Hope this helped!
• What happens if you have a denominator that happens to be a transcendental number? You can't rationalize it, so is there anything you can do with it?
• It is perfectly fine to leave transcendental numbers in the denominator.

Actually, rationalizing the denominator was important in the days when there were no calculators and you had to look up the values of irrational numbers in a table and then do the math. Since multiplication by hand is easier than division by hand, especially when dealing with irrationals, which need lots of digits to maintain accuracy, the practice of rationalizing the denominator (to put the irrational number in the numerator) was developed. Even though we have calculators now, it is considered good form to always rationalize denominators when you can.
• How do you solve rationalize a denominator when there are two radicals? An example would be 1 / (1 + √3 - √5)
• Good question Malachi,
There may be an easier way but the way I figured it out takes two steps because of the three term denominator.
1) Switch the plus and minus signs of the denominator then multiply giving you
1 / (1 + √3 - √5) * ((1 - √3 + √5) / (1 - √3 + √5))
2) After distribution, the denominator simplifies to
-7 + 2√3√5
so the fraction we have so far is
(1 - √3 + √5) / (-7 + 2√3√5)
3) We still have radicals in the denominator so we repeat step 1
((1 - √3 + √5) / (-7 + 2√3√5)) * ((-7 - 2√3√5) / (-7 - 2√3√5))
4) The denominator simplifies to -11. The numerator is kind of gnarly but the entire fraction now looks like:
(-7 - 3√3 - √5 - 2√3√5) / -11
You may want to double check my math but I'm pretty sure I copied my paper and pencil notes correctly.
Hope this helps!
• why √2/2 doesn't equal 1/2?
• I'm assuming that you mean (√2)/2. This is because the ­√2 does not equal 1. √2 is an irrational number, which means it goes on forever without repeating past the decimal (the first 10 digits of √2 are 1.414213562). So (√2)/2 would be half of 1.414213562 which is about 0.707.

√(1/4) is equal to 1/2, however, as is (√1)/2.

If you're asking about √(2/2) though, that would be 1. Since 2/2 is 1 and the √1 is just one.
• Do you have to rationalize the numerator if it is irrational like √2?
• No, it's okay to have an irrational in the numerator. It's only when an irrational is in the denominator is when you have to get rid of it.
• Why do we multiply by 2+√5/2+√5 instead of 2-√5/2-√5? If we're multiplying numerator and denominator by the denominator, shouldn't it be 2-√5?

I understand that (a-b)(a+b)=a^2-b^2, but it's not fully clear why we use 2+√5 to rationalize the denominator, since that's one of the two terms used to obtain a^2-b^2.

Would appreciate anyone shining a light on my confusion.
Thank you!
• The goal of rationalizing the denominator is that we want no radical in the denominator when done. Multiplying (2-√5)(2-√5) does not eliminate the radical in the denominator. You can see this as you multiply it out:
(2-√5)(2-√5) = 4 - 2√5 - 2√5 + V^2 = 4 - 4√5 + 5 = 9 - 4√5

We use (2+√5) because it creates a difference of 2 squares in the denominator. By creating a difference of two squares, the radical does get eliminated.
(2-√5)(2+√5) = 4 - 2√5 + 2√5 - √5^2 = 4-5 = -1

Hope this helps.