Properties of Exponents

When working with exponential notation, it is useful to understand some of the basic properties of exponents. This is probably review for many of you, but I will post this here now for those who are just learning it for the first time or need a bit more help to understand it.

First, let's examine the form of exponents.

If we have a real number, a, and a natural number, n, we can define the exponential expression aⁿ as:

aⁿ = a x a x a ... a (n factors of a), 

where we say that a is the base and n is the exponent. The exponent is also frequently referred to as the power.

Now, knowing the basic form of an exponential expression, let's look at some of the properties of exponents that will prove to be very helpful to you in the future!  These will be used over and over again, and will be the basis of future lessons, so it will be very good for you to understand these well.  There are really only four of them, so it shouldn't be too hard to memorize them.

1.  The first property defines how to multiply exponents.  If we have a real number, a, and natural numbers n and m, we can say:

a^maⁿ = a^m+n

So, what this basically says is that when you have exponential terms to multiply together, if they have the same base number, you can simply ADD the exponents.  To demonstrate this, we can write out an example.  If we want to simplify a²a³, we can expand it to show [a x a] x [a x a x a], which is a x a x a x a x a, which is a⁵. The base a was common, so we could just add the 2 and 3 in the exponent.  Same thing applies when the terms are bigger.  Try for yourself to prove that (x+1)²(x+1)⁵ = (x+1)⁷.

2.  On the other hand, a similar property exists to divide exponents.  If you want to divide exponential terms, you just subtract the exponents, like this:

a^m / aⁿ = a^m-n

Prove to yourself that a³ / a² = a, in the same way I demonstrated above.

3.  If we want to raise an exponential term to another power, such as with (a^m)ⁿ, we can simplify it to a^mn.  Again, try it yourself by writing it out like I did above to demonstrate this with the expression (a⁵)² = a¹⁰.

4.  Finally, similarly to multiplying a coefficient times everything inside a set of brackets, if we want to raise a product or quotient expression by a power, the exponent can be written for each term, like so:

(ab)^m = a^mb^m, or

(a/b)^m = a^m / b^m

In this case, it should be simple to see that this means something like (ab)² = a²b². If you want to start combining these properties, you can see something like (a⁵b²)² = a¹⁰b⁴.

Hopefully with these examples you can see that the properties of exponents are not that complicated to work with, and once you get the hang of the basics and can start combining them, they are actually very easy to use!

Polynomials in Mathematics

In this post, I would like to briefly refresh the concept of Polynomials in Mathematics, as at the start of the school year there are always students looking for extra polynomials help.

I would like to explain some basic terms and definitions that will hopefully make working with polynomials less confusing.

The first two terms which are fairly obvious are "variable" and "constant".  You probably don't need me to tell you that constants are terms that do not change (they are fixed... they don't always have to be known, but they do not change), whereas you can change the value of the variable to arrive at a different solution to your problem.  As in, if you vary the length of the square, you calculate a different area of the square.  The general practice is to assign letters near the start of the alphabet as constants (a, b, c), and at the end for variables (x, y, z).  These terms are self-explanatory, so I will leave them at that.

Substituting values into your variables only makes sense in some cases.  E.g.  sub in -4 for x, when the expression is square root of x.  It doesn't make sense.  Therefore, the set of numbers that DO make the expression make sense is called the "domain of the variable".

An example of this concept can be seen here.  Find the domain of the expression 1 / (x-5).

You can substitute any number in for x and calculate a solution, except for one.  You cannot have a zero as the denominator, as that makes the expression undefined.  So, in this case, you have to ask yourself "what makes the denominator equal zero?"  You can calculate in this case that x - 5 = 0, and so x=5.  Therefore, the domain of the original expression is all real numbers, except x cannot equal 5.

The term "polynomial" is given when you have a series of "terms" in an expression with different powers of x as long as the powers are not negative.  In general, a polynomial looks like this:

a_nxⁿ + a_n-1x^n-1 + ... a₁x + a₀

Each x value is contained in a "term".  The a values are "coefficients", which are constants.  They do not vary.  Overall, the "degree" of the polynomial is the highest power of x.  General convention is to right the terms from the highest degree of x to the lowest.

Since a polynomial is a generic collection of many (poly = many) terms, specific forms of polynomials have special names.  A "monomial" is a polynomial that only has one term, such as 2x.  A "binomial" has two terms, such as 4x² - 1x.  And a "trinomial" has three terms (guess what it looks like!).

Polynomial terms of the same degree can be combined.

In the expression 3x + 2x + 1, you can combine the terms so that you simplify to 5x + 1.  You can do this with x² terms and higher as well.

There are a couple of special polynomial products that will appear very frequently in your studies.  I will also go over these in my next post so that you can memorize them to apply them easily.  Hopefully, when you see these, you will understand how they work and will not need to look for polynomials help anymore!

Radicals and Rational Exponents

Having discussed the topic of radicals and nth roots in my previous set of posts, we can apply that information to the concept of fractional exponents. These rational exponents are no more difficult to work with, and once you identify just what they are telling you, you can easily apply your rules for exponents to simplify them and solve your equations. Factoring fractional exponents may look tricky, but you will see that they follow the same rules that you already know.

Consider the expression 2^1/3. Knowing your rules for exponents, you can see that if we cube this so that we get (2^1/3)³, this is equal to 2¹, which is just 2. However, what you will realize is that what we have just done is apply property #1 of the properties of nth roots. That is, (ⁿ√x)ⁿ = x. So, what we have just shown is that 2^1/3 is the same as ³√2.

So then, with that example we can now extend what we've done to come to a definition of rational exponents and roots.

Part 1:  If we let b be any real number, and n is a natural number, b^1/n can be defined by ⁿ√b. (if n is even, b must be greater than or equal to 0).

Furthermore, we can extend our definition of exponents fractions to include any fraction, not just 1/x type.

Part 2:  If we let m/n be a rational number, n is positive, and ⁿ√b exists, we can say that b^m/n = ⁿ√(b)^m or (ⁿ√b)^m. (Both forms say the same thing and work the same way.)

It is important to realize and understand that these rational exponents follow the same properties as integer exponents, which you are likely already familiar with (I'll do a post soon to refresh you, or for those who have not learned that topic yet).

So now that you know what rational exponents are, let's take a look at a few examples of how you work with radical and rational exponents.  Simplifying rational exponents takes some practice, but it is all the same math that you have already learned.

1.  Simplify the following expressions:
a) 64^1/2
b) -64^1/2
c) (-64)^1/2
d) 64^-1/2

a) This is simply the square root of 64, √64, which is 8.
b) Recall the notation with negative signs and nth roots. This means -(64^1/2) = -√64 = -8.
c) As there is no such real number that gives -64 when squared, this expression cannot be simplified to a real number.
d) 64^-1/2 indicates √64^-1, and a negative exponent means that you take the inverse of it. So,64^-1 means 1/64, and then the square root of this is 1/8.

2.  Simplify (4x^2/3)(3x^1/4)
(4x^2/3)(3x^1/4) = 12x^{(2/3) + (1/4)} (by the exponent rules)
= 12x^11/12

3.  Simplify (√x)(³√y⁴)
= (x^1/2)(y^3/4)
Alternatively, if we give the fractions a common denominator, we can rewrite as follows:
(x^1/2)(y^3/4) = (x^2/4)(y^3/4) = (⁴√x²)(⁴√y³) = ⁴√(x²y³)

So, there you have it.  Working with radicals and rational exponents is not that different from working with any other exponents, once you know what you're doing.  Go over my examples again to be sure you understand the concepts, and practice some more on your own, and you will be an expert in no time. Drop me a line in the comments if you have any problems.

One final note: irrational or radical exponents follow the same rules as well, and I will cover some of those examples in a future post.

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.