The Root of All Math Puns

Fucking exponents, how do they work???

Actually this post is about exponents AND roots, because - get ready to have your mind blown - they are the same thing.

WHOA. BACK THE FUCK UP, GUINDO. What do you mean ROOTS and EXPONENTS are the same thing?

I proved addition and subtraction were the same fucking thing and you STILL doubt me? Sheeeeesh you people are ridic.

Okay, before we go any farther let me explain what I mean by roots and exponents. Exponents are those neat little superscripts like the 2 in x². Roots are square roots, cube roots, and so on, that weird looking long division symbol over a number: .

What an exponent means is "multiply this by itself this many times." 2² means 2*2 and 2³ means 2*2*2, and so on. This is also called "power of [number]." Squares are powers of two, cubes are powers of three, etc.

A root means the opposite. What times itself this many times gets you this number? means "what times itself three times gives you 8?" (Hint: the answer is 2.)

Roots and exponents are inverse functions. That means if you take the square root of something squared, both the root and the exponent cancel out. BUT WHY DOES THAT HAPPEN? It really isn't "because I said so," I promise.

The first thing that needs to be explained is this: a root is really a fractional exponent. That's right, . Likewise, , and so on for whatever number you might put on top of that little spike (what the fuck is that thing even called anyway).

The second thing is this: when you take something with an exponent to an exponent, you multiply both exponents. When you take the square root of both sides of an equation, what you're really doing is taking both sides to a power of 1/2. So what does that mean in practice? Let's check it out.



As you can see, the exponents end up multiplying out to become 1, which means both your root and your exponent can drop out.

You can also use this knowledge to re-write stuff into a doable form. You can't really take something to the 3/2's power without a calculator, but you can take the square root of the cube of something.



First you split up the exponent into ones you know how to do (a cube and a square root), and then you calculate.

Pretty cool! I think. And my opinion is the only one that matters.

>=|


BUT WAIT! THERE'S MORE!

("There's MORE? Auuuugh.")

Silence, hapless reader. There are RULES about exponents. These rules are pretty dang strict and if you break them YOUR ANSWER WILL BE WRONG FOREVER. So let's talk about exponent rules.


You CAN NOT split up an added quantity with an exponent. This is not true and don't do it ever. If you have multiple terms inside a quantity taken to an exponent, you HAVE to either get rid of the exponent algebraically or expand the quantity by multiplying it out using the definition of an exponent (which is a thing you can do and I will cover it eventually).


On the other hand, you CAN do this with multiplication. What this rule is really handy for is reducing roots. If you end up with, say, , you can reduce it using this rule. Let me show you how:




Expanding on the multiplication rule, it applies to division as well. Which shouldn't surprise you because division is the same as multiplication.


If your bases are the same (and ONLY if your bases are the same, DO NOT TRY THIS IF YOUR BASES ARE DIFFERENT), then you add the exponents together. Shove the following into your calculator to confirm if you're curious:





And the one we've already discussed, if you have exponents to an exponent, then you multiply them.

But gosh, Guindo, you may ask. What about adding stuff with exponents?

Here's an unfortunate truth. You can only add stuff with exponents if that stuff has the same exponent. You cannot, CAN NOT, add x³+x². Can't do it. On the other hand, x²+x² CAN be added, and it is 2x². The exponent doesn't change. You can plug in some values for x and test it out if you don't believe me.

Next time, oh, I don't know, let's talk a little bit about using letters and shit to stand in for numbers. NEXT TIME: VARIABLES!