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Rationalizing the Denominator of a Radical Expression (Slice: Radicals)
Rationalizing the denominator of a radical expression means to make the denominator a rational numberone without a radical.
Recall that a rational number is any real number that can be written as a ratio of two integers. An alternate definition is that it can be written as a decimal that ends or repeats. If a number is irrational (not rational), it must be the opposite of a rational. Hence, you cant write them as ratios of integers and if you look at their decimal forms, the decimals dont end but they dont repeat either. The most famous irrational number is probably Pi. The most common irrational numbers you deal with are square roots of numbers that arent perfect squares.
If you put into your calculator it will display as many digits as it canget a bigger calculator you get more digits because the decimal form of this number never ends. In mathematics we often need to deal with exact numbers not rounded off versions form your calculator so you get to learn all these techniques for simplifying radicals because there isnt any way to write them as exact decimals.
There are rules on what form to leave a radical just like we have rules that tell us when a fraction is reduced. Two things to remember about radicals
1. You can leave a perfect square factor under the radical symbolThat would be like not reducing 2/4 to . Thus, is the proper way to leave this radical.
2. You cant leave a radical in the denominator of a fraction.
So how do you get rid of a radical????
Recall that you can multiply the numerator and denominator of a fraction by the same number and not change the value of the fractionas long as you do the SAME thing to both numerator and denominator. This is the same thing as multiplying the fraction by 1.
For example: Note I multiplied top and bottom by 3, but the value of both fractions is still 2.
We use this same idea to rationalize denominators. Look at the denominator and figure out what you should multiply it by to get a perfect square under the radical. If you dont know the perfect squares this is not going to be easy. (2, 4, 9, 16, 25, 36, 49, etc.)
Finally an example:
Rationalize the denominator and simplify:
I have a couple of options here. I can simplify the denominatornotice it has a perfect square factor, but this is still going to leave a radical in the denominator so Im just going to deal with that issue first and then reduce what I end up with. It won't matter if you reduce first and then rationalize...you still get the same answer.
I need a perfect square on the bottom. The closest one that has 8 as a factor is 16. I can get a 16 under the radical in the bottom if I multiply that 8 by a 2. Since the 8 is under the radical the 2 must be also. You can multiply or divide numbers that are both under the radical or both outside the radical but you cant do anything if one is under and the other isnt. This is easier to understand the number you see under the radical is not that number. A is not a 3. It is the number when multiplied by itself would give you a 31 point something that never ends. Check it out on your calculator. It isn't a 3 so don't treat it like one!
Back to the problem.
We decided to multiply the bottom by the square root of 2 and whatever we do to the bottom we have to do to the top so the value of the original fraction remains unchanged. (Check on your calculator that all 3 fractions below give you the same decimal.)
(Notice I am really multiplying the original fraction by 1 because )
You cant reduce the 6 and 4 because the 6 isnt really a 6. It is the number that multiplied by itself is 62 point something that never ends.
Example 2:
Since both numbers are under the radical, I can reduce the fraction by dividing a 5 out of both the numerator and denominator (really a square root of 5). I could rationalize first and reduce later, but this will make my numbers smaller and easier to work with.
I still need to rationalize. The closest perfect square that has 3 as a factor is 9. I can get a 9 in the denominator by multiplying by a square root of 3. Remember, whatever I do to the top, I must do to the bottom. That makes the multiplier a 1 and multiplying by 1 doesnt change the value of the number you are multiply.
Again we cant reduce further because the 33 is under the radical (it isnt really 33) and the 3 is not.
Example 3:
We can get a perfect square in the denominator if we multiply by the square root of 6.
This time we can reduce because we have a 2 and a 6 outside the radical. We reduce just like any other fraction without a radical in it 2/6 reduces to 1/3 and the final answer is
(No need to write the 1 factor in the numerator.)