Rationalizing Denominators

In my previous posts, about nth roots and the properties of nth roots, I mentioned that I would post next about rationalizing denominators, and how to rationalize the denominator.  This may be a new concept to some students, so I will explain it first and describe where it will come in handy, before I actually show you how to do it.

Rationalizing denominators is a way of simplifying radicals, specifically fractions that have a radical on the bottom (in the denominator).  You may come across expressions like these, for example, when dealing with property #3 of nth roots.  In fact, you will soon find that equations are often not considered 'completely simplified' if there is still a radical in the denominator.  To get rid of this, you must rationalize the denominator.

Here's how you do it.  I think you will find it much easier than its name makes it sound!

First, let's consider a couple of points.

The first one is an obvious one, and you will see why in a second... that is, x / x = 1.  Right?  Anything divided by itself is 1 (except zero).  3 / 3 is 1.  100 / 100 is 1.  57483 / 57483 is 1.  That's how fractions work: if you have all of the pieces of the pie, out of all the pieces of the whole pie, you have the whole pie.  Easy, probably doesn't even need to be said.  But keep it in mind.

The second point is a refresher of the properties of nth roots... specifically, property 1 which says (ⁿ√x)ⁿ = x.  So, for n = 2, we have (²√x)² = (√x)² =  (√x)(√x) = x

So then, with those two points in mind, here is the trick to rationalizing the denominator when there is only a radical down there.  Quite simply, you take the expression that you have, and whatever radical is in the denominator, you multiply both the top and the bottom by that.  Let me show you what I mean:

Take 2 / √5.  To rationalize the denominator, we multiply the top and the bottom by √5.  So, this is what we get:

Hopefully, you can easily see how we really haven't really changed anything at all.  We multiplied the whole thing by 1, and then simplified the top and bottom using simple multiplication and division rules that you already know.  And that would then be considered to be fully simplified.  However, I should point out that this is all that's necessary when there is ONLY a radical in the denominator and nothing else.

If we had √5 + 2 in the denominator, that needs a bit more work.  It's still rationalizing the denominator, because we 'make the denominator rational' by getting rid of the radical.  But, when there's a plus or minus down there as well, there is an extra twist.

We still multiply the whole thing by 1 (that is, by something over something), but now it's a bit more complicated.  If you try to multiply top and bottom by √5 + 2, YOU ARE INCORRECT.  You can try (just follow the typical FOIL rules for multiplying binomials), but you will find that you end up with more of a mess on the bottom... more terms than you start with!  It doesn't simplify anything at all!

To properly get rid of this radical, you multiply the top and the bottom by √5 - 2.  That's right, notice that the sign in the middle is flipped!  This term is called a conjugate, and the two terms with the opposite sign in the middle are conjugates of each other.  If you FOIL out (√5 + 2)(√5 - 2), you end up with 5 - 4 = 1 on the bottom!  (It won't always be 1, but you will always get rid of the radical and can simplify further!).

So, let's try another example to put it all together.

Rationalize the denominator and simplify:  5 / (2 + √6)

And that would be your simplified answer!  Note that I multiplied the top and bottom by -1 (still not changing the equation!) to switch the negative signs around so that I didn't have a negative on the bottom.  Not always necessary, but it looks much cleaner.  (Technically, some teachers would say the proper answer is (5√6 / 2) - 5.  That works too.)

So that is all there is to rationalizing the denominator, and using conjugates to do it.  Since this is now a fairly long post, I'll leave it there.  But if you'd like another example or two with variables instead of numbers, or nth roots higher than squares, leave me a note in the comments and I'll either update the post or do another one with more!

Properties of nth Roots

Having discussed the concept of nth roots in my last post, here I want to show you some very important Properties of nth Roots that will help you to understand and solve you homework questions with nth roots.  There are only a few, and I think you will find that they seems pretty easy.  It's good for you to see them though, and realize that there are specific properties and rules that apply to nth roots.

Before we get into it, you may or may not have been taught this already, but I will briefly mention it here because I think it may help clear up some confusion with the rules of nth roots.  When you have a root, you can express that as an exponent and it means the same things.  And you do that by considering the index to be a fraction exponent.  Probably not a very good choice of words, but this should be clear with this short demonstration:

²√x = x^1/2

If you get confused or muddled up with the following properties and rules, try thinking of them in terms of this example, and then you should be able to apply the rules of exponents that you already know that help you sort things out.  Square root notation is just a fancy, specialized way of working with fractional exponents.

So, here we go.

To start with, we have to state some assumptions and generalizations.  We let x and y be real numbers, and m and n are natural numbers. In this case, the following properties are true as long as the expressions are not undefined:

1. (ⁿ√x)ⁿ = x
Like I said, most of these make sense if you think about them for a second.  In this case, consider that most (if not all) arithmetic functions have opposites.  Addition is opposite to subtraction, multiplication to division.  In this case (let's assume n = 2 for a second) square root is opposite to squaring.  Or, cube root is opposite to cubing.  And so on.  Square root of 16 is 4.  4 squared is 16.  They cancel each other out, and you're left with x.  Makes sense?  E.g. √x² = x

2.  ⁿ√xy = ⁿ√xⁿ√y
Again, this one kinda makes immediate sense.  Kind of like multiplication of a polynomial, you multiply everything inside the brackets by what is on the outside (ie.  2 x (x + 3) = 2x + 6 ).  In this case, the root is applied to everything beneath the root sign.  It is important to make sure that you realize that this law holds only for multiplying (and dividing... see #3) terms beneath the root sign, but does not work for adding or subtracting terms.  So, ³√xy = ³√x³√y

3.  ⁿ√(x/y) = (ⁿ√x) / (ⁿ√y)
This is a follow-up to #2.  As long as the terms beneath the root sign are multiplying or dividing, you can re-write the expression like this, with a radical sign on the top dividing by a radical sign on the bottom.  Again, this doesn't apply if you are adding or subtracting x and y beneath the radical sign.

4. a) if n is even:   ⁿ√xⁿ = |x|
This is connected to how you can square a positive or negative number and get the same POSITIVE result.  So, if you try ²√(5)² you get positive 25.  Same thing if you try -5... positive 25.  If the n term is even, you will always get the positive, or ABSOLUTE VALUE, of x.  However...

    b) if n is odd:   ⁿ√xⁿ = x
In this case, you CAN finish with a negative number.  If you try  ³√(5)³ you get 5, but if you try ³√(-5)³ you get -5.  Write it out long-hand to see why.  -5 x -5 x -5 = -125.  Cube root of -125 is -5.  There is only one answer.

5.   ^m√(ⁿ√x) = ^mn√x   (Sorry, this one is hard to type out)
Once again, this one is kind of like multiplication, and the rules of exponents.  If you have (x,²)², the rules of exponents say that you can multiply the exponents to have x⁴.  In this case, the numbers are in a different place, but the same kind of rule holds.  ²√(³√x) = ^2x3√x = ⁶√x.

You will likely find most use of these properties in simplifying larger expressions, making them easier to work with.  Such as, pulling out perfect square or cube from a larger number:
E.g. √72 = √9√ 8 = √9 x √4 x √2 = 3 x 2 x √2 = 6√2

In my next post, I will explain a concept that always seems to give students difficulties... conjugates, and rationalizing the denominator.

nth Roots

This brief post is going to explain to you what nth roots are.  Following posts will show you how to work with them in your equations.

Going back to elementary algebra lessons, you were taught the concepts of square roots.  For example, the square root of 25 is 5.  This notion was explained, but likely never generalized much past saying "what times what gives you your number?"  However, like many things in the mathematics world, there is far more to roots than this.

Enter the concepts of "nth roots".

Everyone knows the symbol for square root, and that it means "what times what gives the number underneath this symbol".  Some of you may have seen a tiny 3 written on the top left just above the root sign.  What this means is a touch more complicated: "what times what times what gives the number underneath this symbol?"  It is called the cube root.

It's always simpler to go forwards before going backwards, so let me show you one:

2 x 2 x 2 = 8

You can say 2³ (2 to the power of 3) equals 8.  Straight-forward, right?  Now, go backwards.  Find the cube root of 8.  Well, in this case, "what times what times what" is "2 x 2 x 2", so the cube root of 8 is 2.

So then, we now understand square roots, and cube roots.  Now, naturally, we can expand on those.  In general terms, nth roots can be defined as this:

a^n = b     (a to the power of n equals b)

and therefore a is the nth root of b,

where n is a natural number, and a and b are real numbers.

Some examples should hopefully clear up the complexity of what I just wrote.

3 and -3 are square roots of 9, because 3² = 9

4 and -4 are fourth roots of 64, because 4⁴=64

2 and -2 are sixth roots of 64, because 2⁶=64

Further, what these examples should demonstrate to you is that all EVEN roots of postive numbers occur in pairs, with there being one positive and one negative number for each.  Therefore, to distinguish between the two, there has been accepted a common notation:

The positive, or principal, nth root is designated as the root sign with the nth root number over it:   ⁿ√
E.g. ⁴√16 = 2, and NOT -2
To indicate -2, you write the initial expression as -ⁿ√
E.g. -⁴√16 = -2

On the other hand, all ODD roots only occur singly.  Such as 2 is the fifth root of 32, whereas -2 is the fifth root of -32.  (Write it out to see the differences.  -2 x -2 x -2......)

A little bit more nomenclature, just so you always know what is going on now:
What you have always knows as the "square root sign" is technically called the radical sign.
The number underneath the radical sign is called the radicand, and the number n used to indicate the root is called the index.

To summarize all of this, we can revise our original definition to this:

If n is a natural number, and a and b are non-negative real numbers, then

ⁿ√b = a, if and only if b=aⁿ

^{The number a is the prinicpal nth root of b}

Also:

If a and b are negative and n is an odd natural number, then

ⁿ√b = a, if and only if b=aⁿ

^{So there you have it.  You now know what is meant by nth roots.  In my next post, I will go over some of the properties of nth roots that will make your homework questions a lot easier.}