Fact-orials

Monday, January 21, 2019

Roots

Background

We've covered what an exponential is and how it works. We can also go the other direction...

Question

Evaluate:

Answer

Analysis

Before we dive into what a root is and how it works, let's start with an exponential. Remember that we can take a number, say 2, and multiply it with itself a certain number of times, say 2. That looks like this:

We can go the other way, starting with 4 and working our way back to 2 (we'll use exponential rules to do it):

Let's look at the blue terms above, and in particular the

. What is this?

It's called a root and is the inverse relation to an exponential.

There are various ways to express the operation of taking a root. One is to use the exponential form I used above - where we use an integer to express taking something to an exponential power (so the power 2 says we take a number and multiply by itself twice), we use a fractional exponent the root we want to take over 1 (and so the fraction 1/3 in the exponential position means we want the number that when multiplied by itself 3 times gives us the given number).

Another way to express a root is to use the symbol that looks sort of like a dividing sign (below is the square root of 2):

and the cube root of 27:

Notice that if there isn't a small number sitting in the "crook" of the root sign, assume you are taking the square root, or the exponential fraction 1/2.

Now let's talk about some things about the numbers that result from taking a root.

Let's start with even roots.

Notice that if we look at the square root of 4:

I'm looking for the number or numbers that, when multiplied by itself twice, gets me to 4. What numbers might they be?

One is 2: 2 x 2 = 4. Great! There is another number I can use, though, to get to 4: -2 x -2 = 4. And so there are two answers when I take the even root of a number - one will be positive and the other negative. Because of this, sometimes people will express the root as an absolute value:

When taking an even root, this will always happen!

This also means that if I take the even root of a negative number, I won't end up with a real number. For instance, if I take this number:

there are no numbers I can multiply by itself to get to -4. (To get around this, mathematicians use Imaginary Numbers - that'll be in further entries).

So what happens if I take an odd root?

Let's look at the cube root of -27:

If I multiply 3 x 3 x 3, I get 27. Which means that the cube root of 27 is 3.

So what happens if I take the cube root of -27?

Notice that I can say -3 x -3 x -3 = -27, and so therefore the cube root of -27 is -3.

Whenever we take an odd root, whatever sign the number under the root sign is the sign of that numbers root.

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