(September 29, 2018 at 1:57 am)Reltzik Wrote:(September 26, 2018 at 8:06 pm)Aliza Wrote:
Others have already answered how this is done, so I'm going to try to translate it into ordinary-people-speak as best as I can. I'm going to err on the side of dumbing it down, and that will likely sound patronizing, and I will further err in the direction of exaggerating that sound for the sake of humor. Please know I mean no offense in doing so.
Let's start by reviewing the basic definition of an exponent. Yes, I know you already know this, but like I said, humorous exaggeration.
25 means "multiply five instances of the number 2 together", or 2*2*2*2*2 = 32. Similarly, 23 means "multiply three instances of the number 2 together", or 2*2*2=8.
While we're defining stuff we already know (fun!), let's define multiplication. 2*5 means "add five instances of 2 together", or 2+2+2+2+2=10. (We can also reverse the order and have it be "add two instances of 5 together", which we can't do with exponents, but that's not important here.)
We can reverse these definitions as well. So for example, when we see 2*2*2*2*2, we can rewrite it as 25, and when we see 2+2+2+2+2, we can rewrite it as 2*5.
Now, let's look at your problem: How do we show that 2x+1+2x+1 -2 and 2x+2-2 are equal? I'll apply justifications/expansions for each step each in their own paragraph labeled by a letter, and then put all that together in one final proof.
A) Let's start by looking at 2x+1+2x+1. HEY! What's that? We're adding something to itself! That's multiplication! We have two instances of 2x+1 added together! That's 2*2x+1.
B) Can I write out 2x+1? Sure... kinda. I can write it out as 2*(2*2*...*2), where I know that there are a total of x+1 2s within the parentheses. The notation is difficult and awkward, but the math that the notation represents is perfectly legit.
C) But wait! Multiplication is associative -- that means that the order of parentheses don't matter. That means that we can just write it as 2*2*2*....*2, where there are a total of x+1 2s that were originally within the parentheses, and one additional two that was originally in front of the parentheses. That's a total of x+2 instances of 2 multiplied together.
D) Which in turn means we can write it as 2x+2
E) You asked about a general rule. It's not a factoring or distribution rule as you thought, but an exponent rule. Steps B, C, and D can be combined together in a general rule as follows: a*ak=ak+1. Or in other words, if you multiply an exponential expression like ak or 2x+1 by yet another instance of its base (a or 2, respectively), it's the same thing as adding 1 to the exponent, and adding 1 to the exponent is the same thing as multiplying yet again by the base. This is a very basic, very important rule for exponents -- so much so, that it pretty much defines what an exponent IS.
So laying all this out:
A) 2x+1+2x+1 -2 = 2*2x+1 - 2
E) 2*2x+1 - 2 = 2(x+1)+1 - 2 = 2x+2 - 2
Notice by the way that the -2 wasn't really all that important. You had to keep it there because it changed the values of stuff and so was significant, but it wasn't part of any mathematical tricks leading to the answer. It was just a red herring designed to force you consider, weigh, and discard a bunch of possible false approaches, such as factoring, before settling on a true approach. We math folk are mean meanie-pants like that.
Going back to your own attempt, you make an error on the very first line: You can't add bases like that! If you ever doubt that a rule you're trying works, just plug in a few numbers and see if math turns screwy. If it doesn't, you're probably okay. If it does, your rule isn't actually a rule and you can't trust it. For example, adding the bases would say that 2x+1+2x+1 = 4x+1. Does this work? Well let's try plugging in 1 for x. We'd get 2(1)+1+2(1)+1 = 4(1)+1. Evaluating this reduces it down into 8=16. Obviously something's wrong with this! When you factored out the 2 in the next line, that was another mistake.... which weirdly managed to make everything right again! (Math mistakes almost never do this.)
We can think of addition, multiplication, and exponents operating on different "levels". Each of these operations mixes easily with stuff on the same level -- the rules for combining addition with more addition, or multiplication with more multiplication, are very straightforward. Same with their inverses -- you can mix multiplication and division up very, very easily. Things get a little uglier with exponentiation, but it's when you start mixing things from different levels that things get ugly really fast. Instead of nice neat rules like commutative and associative rules, which are only hard to remember because you wonder why we even needed to bother spelling them out at all, we instead have ugly rules like the distributive law. And that's when we're lucky. Often times not even the distributive law can be used, and we're left with no tools at all that do what we want.
So the moral of that last paragraph is this: Addition, multiplication, and exponentiation don't mix easily. Whenever you can, stick to just one of these types of operations. When you can't, brace yourself for ugliness. If you don't know what to try first, try applying exponent rules to exponents and multiplication rules to multiplication and so forth... not because that's always or even usually the right answer, but because it's the easiest option to check.
Sorry again this sounded patronizing. I was just trying to (humorously) err on the side of comprehensibility.
You're not coming across as patronizing at all! I'm *just* about to turn in for the night, but I wanted to thank you in advance for writing it. I'll really read it and dissect it tomorrow.
