Help! Exponents are too hard for me

[GrandizerII]

Forget the exponents in that step. You have 4 - 2. What's the common factor? 2.

2 is the same as 2^1, so it's written that way in order to emphasize the exponents. That's all. Look at it as simple factoring, with exponents added in after the factoring out.

[vulcanlogician]

YO GRANDIZER! Nice to see ya, man.

[Aliza]

Oh, I see now!!!

OMG! That was so much more basic that I was making it out to be.

Thank you, thank you, thank you to each of you that replied!

[vulcanlogician]

*Smugly dusts off his cuffs*

We all know it was my post that solved that one

[Aliza]

*Smugly dusts off his cuffs*

We all know it was my post that solved that one

If not for you, vulcanlogician, I'd have been banging my head on the desk for another 8 hours. Thank you.

[Kernel Sohcahtoa]

Oh, I see now!!!

OMG! That was so much more basic that I was making it out to be.

Thank you, thank you, thank you to each of you that replied!

Aliza, just so you know, IMO, you sound like a good mathematics student.  It is completely normal to not be able to immediately wrap one's head around a mathematical concept.  In particular, it often takes multiple readings/attempts combined with fumbling around with problems in order to gain a solid understanding of the material and an appreciation for it.  It sounds like you've just accomplished that, so well done. 

P.S. Good luck with your studies, and I hope that you enjoy the material.

[Fireball]

Forget the exponents in that step. You have 4 - 2. What's the common factor? 2.

2 is the same as 2^1, so it's written that way in order to emphasize the exponents. That's all. Look at it as simple factoring, with exponents added in after the factoring out.

Grandizer! Welcome the fuck back!  Your math thread has cobwebs all over it. :looking askance:

[Reltzik]

---

Hello AF mathy people! You know who you are. I'm stumped on how this expression is simplified. I thought I understood the rules of exponents well enough, but without having this explained to me, I'd have never gotten there... and even with the explanation, I was still unable to really grasp how it was done.

I'm trying to push symbols around to get this:   2ˣ⁺¹ + 2ˣ⁺¹ - 2    into this:   2ˣ⁺² - 2

Add like bases (keep the same exponent)

4ˣ⁺¹ - 2

Factor out the 2

2(2ˣ⁺¹ - 1)

Take note that the 2 on the outside is really 2¹

2¹(2ˣ⁺¹ - 1)

Now re-distribute the 2, so now I'm multiplying 2¹ by 2ˣ⁺¹. So that's same base, add exponents. There's my x+2, and I've arrived at the form I need this in.

2ˣ⁺² - 2

Question: What magical exponent rule enables me to factor out the two and multiply it back in to change the exponent? Would any math people here have intuitively known to do that?

---

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.

---

... oh.

... there was a second page to this thread.

... in which Aliza had already understood everything.

.....

...

.

[Aliza]

---

Hello AF mathy people! You know who you are. I'm stumped on how this expression is simplified. I thought I understood the rules of exponents well enough, but without having this explained to me, I'd have never gotten there... and even with the explanation, I was still unable to really grasp how it was done.

I'm trying to push symbols around to get this:   2ˣ⁺¹ + 2ˣ⁺¹ - 2    into this:   2ˣ⁺² - 2

Add like bases (keep the same exponent)

4ˣ⁺¹ - 2

Factor out the 2

2(2ˣ⁺¹ - 1)

Take note that the 2 on the outside is really 2¹

2¹(2ˣ⁺¹ - 1)

Now re-distribute the 2, so now I'm multiplying 2¹ by 2ˣ⁺¹. So that's same base, add exponents. There's my x+2, and I've arrived at the form I need this in.

2ˣ⁺² - 2

Question: What magical exponent rule enables me to factor out the two and multiply it back in to change the exponent? Would any math people here have intuitively known to do that?

---

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.

[Aliza]

Sorry again this sounded patronizing.  I was just trying to (humorously) err on the side of comprehensibility.

Your explanation is beautiful and I mean that from the bottom of my heart. If you and I ever get to talking off the forum and I mention what my background is, you’ll know that I said that with absolute sincerity. If I had seen that explanation from the start, I’d have understood immediately. I needed it to be put in context of something that made more sense to me. Of course you can’t change the base of an exponent.

I’ve never done math induction and, for example, we were told to disregard the leading portion of the LHS in our answer, so in my answer, I’m writing something like “….+blah, blah = 2(blah, blah) to the k+1”. I didn’t understand that the LHS isn’t supposed to equal the RHS, but rather that the whole expression is supposed to prove that the equation is true in different cases of k. At first, all I saw was that the LHS didn’t equal the RHS, and my Aliza-brain saw this as breaking the rules of math, and therefore, it must be that no math rules apply. –I have since been sorted out.

I’m in this class where I’m seeing stuff from pretty much every math class I’ve taken to date, and some stuff from classes I’ve craftily avoided. I’m pushing my brain to learn new stuff on the fly and try to remember stuff I haven’t touched in a long time. Plot the complex number in the complex plane and write it in polar form. Lol! Easy! Find the complex conjugate of 6-4i? I knew it had something to do with a sign change, but I expected a little more complexity to the question given that it was sandwiched between two other questions from back in Calc 2. (punchline for casual browsers: it’s 6+4i). ... maybe that's the whole point my professor is trying to drive in. Just look at the question and answer it for what it is rather than approaching it with some expectation.

I just want to say again, Reltzik. I really appreciate your reply. It was very professional and sensitive.