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.
Vocabulary used:
For more information check out these links (comment to add your favourite link):
Where might you have come from?
Fact-orials Index
Operations:
Where might we go?
Numbers:
- Imaginary Numbers
- Exponentials and Rational Numbers
