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!  :)

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!