Square Roots - Part II (The Irrational Number)

Picking up where I left off in my previous post (Square Roots - Part I), I'm going to briefly explain irrational numbers. (Also be sure to check out Part III - Factoring Square Roots!)

A rational number is one that can be expressed as a quotient of two integers (ie. as a fraction). So, all fractions are rational. And all whole number integers are rational (since they can be expressed as "something over one").

Conversely, irrational numbers can NOT be expressed as a fraction. The number pi is commonly given as an example of an irrational number, since it cannot be written as a fraction, and the numbers after the decimal point keep going and going. Going back to my last post, the square root of 24 is also irrational. If you punch it into a calculator, you will see that it is 4.8989794855663....... On this note, you can say that the square roots of non-perfect squares are irrational.

Now, to find the square root of a non-perfect square by hand, you just do the same trial and error method we learned when I explained square roots in my last post, only this time, as you narrow your guesses down, you can use numbers with more and more decimal points. Observe:

Find the square root of 24:
4 x 4 gives 16....
5 x 5 gives 25...
4.5 x 4.5 gives 20.25
4.9 x 4.9 gives 24.01.... needs to be a bit smaller
4.89 x 4.89 gives 23.912.... needs to be a bit bigger, but less than 4.9
4.898 x 4.898 gives 23.9904.... bit bigger still... but still less than 4.9
4.8989 x 4.8989 gives 23.999221.... even a bit bigger, but still less than 4.9

As you can see, this can go on and on, until you have as many decimal places as you want.

Alternately, and more commonly, you would leave your answer as an irrational number, rather than recording decimal places (since technically, writing to so many decimal places, unless you write FOREVER, can be expressed as a fraction.... tenths, hundredths, thousandths, ten thousandths, etc...)

However, you don't want to leave your irrational number in a form that hasn't been reduced yet, do you? Check back to my next post to find out how to factor square roots and express them in simplified radical form.

Square Roots - Part III (Factoring Square Roots)

Where I left off in my previous post about irrational numbers, we were trying to solve for the square root of 24 by trial and error.  Reducing an irrational number can be a tedious job if you do it this way!  Luckily, unless specified otherwise, you are allowed to leave your answer in the "simplified radical form," which you can get by factoring square roots.

"Simplified radical form" is exactly what it sounds like: you simplify your expression and leave it expressed as some radical.  But, you have a little bit of work to do to reduce it.

Factoring square roots is quite simple.  It has to do with factoring perfect squares.  But first, you have to determine the factors of the number under the radical sign, and then if any of those factors are perfect squares, you can pull it (the square root of the perfect square factor) out from underneath the radical sign and put it in front to multiply by it.  That sounds awfully wordy and probably isn't the most concise definition, but I think an example will go a long way to helping you understand factoring square roots, and then leaving them in simplified radical form.

Let's continue with the example from my last post, looking for the square root of 24.  In this case, let's just reduce it to its simplified radical form, and not bother wasting time trying to find the exact decimal answer.

So, the first step is to ask yourself what the factors of 24 are.  (I'm going to ignore the √ for a minute)  For this, you can determine this quite easily by trial and error.  You can find that the factors of 24 are 1,2,3,4,6,8,12,24.  From these now, you want to see if any are perfect squares.  Again, you should be able to easily say that 4 is the perfect square.  So, 24 can be expressed as 4 x 6. That's what we want.  Now, let's look at what we have done so far:

√24 = √(4x6)

Now, as I said before, we can take the perfect square that is under the radical sign, and bring it outside the sign.  This is because of the properties of square roots (property #2), which in this case allows us to write:

√(4x6) = √4 x √6

With that property in mind, it should be a bit easier to see why we are interested in the perfect squares, because now, by factoring perfect squares, we can just rewrite the square root of 4 to 2!  So, we can finally write our expression in simplified radical form as:

√24 = 2√6

I hope this demonstration has explained to you the basics of factoring square roots, and leaving you answers in the simplified radical form.  Let me know in the comments if you'd like another example, and I'll do one for you!  Remember to +1 me if this helped you!  :)

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.}

Perfect Squares

This post should have been put up when I posted about square roots (here, here, and here), because it is the exact opposite of a square root!

Where a square root of a number "x" is some number "y" that, when multiplied by itself, gives "x", a (perfect) square of a number is the result of multiplying a number by itself. That is to say, the square of "y", by multiplying "y" by "y", is "x". You can also talk about "squaring" a number, which is to find what the square is. (It can be both a noun and a verb.)

For example: the square of 4 is "4 x 4" = 16. Also, if you square 4, you get 16.

If you want to think of a visual representation of it, "what is the square of 5" is essentially the same as asking "what is the area of a square with a length of 5?" (Of course, all sides have equal lengths in a square.) As you can probably figure out already, to find the area of a square (or, in general, a rectangle) you multiply the length by the width. So here, it is obviously 5 x 5 and the area, or the square of 5, is 25.

This concept of perfect squares can also be extended to polynomials. For example, let's look at the following case:

(x+1)*(x+1) is the square of (x+1). It can also be written as (x+1)^2. You can do the visual trick i just described above if you want, using x+1 as the side length.

If you multiply (FOIL) these binomials, you get (x^2 + 2x + 1). As it is equal to our original expression, you can also say that this product is a perfect square, just as you can say that 16 or 25 is a perfect square. To find out what the square root of this is expression is, it is the same as asking what the square root of (x+1)*(x+1). Over time you will see patterns and be able to quickly notice that the square root of (x^2 + 2x + 1) is (x+1).

This shouldn't be too difficult of a concept to understand. I will do a few short examples, but post comments if you require clarification:

Find the square of 12:
12 x 12 = 144

Find the square of 25:
25 x 25 = 625

Find the square of (x-1):
(x-1)*(x-1) = (x-1)^2 = (x^2 - 2x + 1)

Find the square of (2x+3):
(2x+3)*(2x+3) = (2x+3)^2 = (4x^2 + 12x + 9)

It can also be noticed, and should be kept in mind, that squares will always be positive. Try it to see: plus x plus = plus... negative x negative = plus. A plus times a negative is NOT a square! Squares multiply the SAME number (sign and all!).

See my previous post about "completing the square" for some more stuff relevant to this post.

Square Roots - Part I

The concept of square roots often gives students trouble. (In fact, it is the topic most searched for on my site!) However, the initial uncertainty and hesitation with this topic is quite unnecessary. Square roots sound daunting, but they're really a simple concept.

The SQUARE ROOT of a number "x" is some number "y", such that when "y"is multiplied by itself, its product is "x". Sounds confusing... but you will see that it's not. The square root sign looks like this: √25 (usually with a line over the top of the number.)  This is called a radical sign, and I will cover radicals in far more detail later.

Example:
Find the square root of 25.

So, going along with the definition I gave above, let's say x = 25. So then, we want to know y... that is, what number, when it is multiplied by itself, will equal 25. In this example, most people will be able to say immediately "5 times 5 equals 25!" And they will be right. The square root of 25 is 5, because when 5 is multiplied by itself, it gives 25.

Now, I'm sure that most of you will be saying something like "That's easy! But what about when the numbers are big... or weird... like 529?" While most calculators can tell you the square root by the touch of a single button, figuring it out by hand can take a little more guesswork. To find the square root of, say, 529 (by hand), you have to just keep trying to multiply numbers by themselves to reach it. Watch:

10 x 10 = 100..... not big enough
15 x 15 = 225..... still not big enough
20 x 20 = 400..... getting closer
25 x 25 = 625..... too big! so we've narrowed it down to between 20 and 25
22 x 22 = 484
24 x 24 = 576
23 x 23 = 529 BINGO!

It takes a bit of work, but you see how it can be done. Now, having seen this, there are a few things to note.

1) While it can be said that the SQUARE ROOT of a number "x" is some number "y" that, when multiplied by itself, gives "x", the SQUARE of a number is the result you get when you multiply a number by itself. So, the square root of 25 is 5, whereas the square of 5 is 25. It is important to understand these definitions and not to mix them up. Pay attention to what the question is asking!

2) The examples and method I described use PERFECT SQUARES. A perfect square is a number whose square root is an integer (whole number). So, the square root of 25 is 5, but the square root of 24 is.... less than 5.... but more than 4 (since the square of 4 is 16). Therefore, 25 is a perfect square, but 24 is not. We would call the square root of 24 an IRRATIONAL NUMBER.

I'll have a bit more to say about squares and square roots in following posts.  Make sure you check out my Square Roots - Part II (The Irrational Number) and Part III - Factoring Square Roots posts!