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?

Don’t have time to address the main question but I don’t get your adding like bases.

Let’s assume x is 1.

According to your post, 2^2 + 2^2 would equal 4^2.

However 2^2 is 4. 4+4 is 8. 4^2 is 16.

(September 26, 2018 at 8:06 pm)Aliza Wrote: 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?

Well, 2ˣ⁺¹ + 2ˣ⁺¹ - 2  does become:   2ˣ⁺² - 2 , but only for the base of 2, not for any other base.  You can't add bases.

Zˣ⁺¹ + Zˣ⁺¹  - 2 

= 2 Zˣ⁺¹ - 2

If Z = 2, then this is 

= 2 x 2ˣ⁺¹ - 2 

= 2ˣ⁺² - 2

(September 26, 2018 at 8:47 pm)Tiberius Wrote: Don’t have time to address the main question but I don’t get your adding like bases.

Let’s assume x is 1.

According to your post, 2^2 + 2^2 would equal 4^2.

However 2^2 is 4. 4+4 is 8. 4^2 is 16.

The full question can be found on PurpleMath. It's the first example and you'll find the relevant component of the equation on the right hand side.

https://www.purplemath.com/modules/inductn3.htm

It's making my head hurt!

I don’t see a 4 in that solution.

In this case however I believe the problem is asking you to prove the general case.

(September 26, 2018 at 9:33 pm)Tiberius Wrote: I don’t see a 4 in that solution.

In this case however I believe the problem is asking you to prove the general case.

Purplemath factored out the 2 at an earlier stage.
= [2^k+1 – 2] + 2^k+1
I added the 2^k+1 to the other 2^k+1 to give me 4^k+1, and then I factored out the 2 from the remaining two terms.

It is a proof. I'm just trying to get from point A to point B, but I can't for the life of me figure out what exponent rule allows for that. Why if I can factor out a 2 and then redistribute it does it not just go back into the equation without affecting the exponent? I understand that (x^2)(x^2) would be x^(2+2) which would be x^4, but why in the case of my example question, did the exponent not change when I factored out that two as such that it would go up by one when I put the two back in?

This is a proof demonstrating mathematical induction, and relies on a much more basic understanding of exponentials. (no offense meant) You can't add things like that. If two things are multiplied, the exponents add, but not if those two things are added. You might have been able to get the right answer in this case, but it's for the wrong reason- it's because it is base two, and not a generic answer because of that. Exponents are not too hard for you, I happen to know that you are plenty smart, in my experience. My recommendation is to find an algebra text (say, College Algebra) that is for College Freshmen and re-work through the section on handling exponentials. That same text will have the details of proof by induction, possibly with a less complicated example.

(September 26, 2018 at 9:49 pm)Fireball Wrote: This is a proof demonstrating mathematical induction, and relies on a much more basic understanding of exponentials. (no offense meant) You can't add things like that. If two things are multiplied, the exponents add, but not if those two things are added. You might have been able to get the right answer in this case, but it's for the wrong reason- it's because it is base two, and not a generic answer because of that. Exponents are not too hard for you, I happen to know that you are plenty smart, in my experience. My recommendation is to find an algebra text (say, College Algebra) that is for College Freshmen and re-work through the section on handling exponentials. That same text will have the details of proof by induction, possibly with a less complicated example.

Check out the link to the purplemath page. https://www.purplemath.com/modules/inductn3.htm. I think I solved it in a very similar way to the way Purplemath did.

Exponents should not be too hard for me, but for some reason, I can't wrap my brain around this. Everyone in class seemed to be solving it like I demonstrated above, and I was thinking, "Derp, derp, derp. Exponents are too hard."

(September 26, 2018 at 9:42 pm)Aliza Wrote:

(September 26, 2018 at 9:33 pm)Tiberius Wrote: I don’t see a 4 in that solution.

In this case however I believe the problem is asking you to prove the general case.

Purplemath factored out the 2 at an earlier stage.
= [2^k+1 – 2] + 2^k+1
I added the 2^k+1 to the other 2^k+1 to give me 4^k+1, and then I factored out the 2 from the remaining two terms.

It is a proof. I'm just trying to get from point A to point B, but I can't for the life of me figure out what exponent rule allows for that. Why if I can factor out a 2 and then redistribute it does it not just go back into the equation without affecting the exponent? I understand that (x^2)(x^2) would be x^(2+2) which would be x^4, but why in the case of my example question, did the exponent not change when I factored out that two as such that it would go up by one when I put the two back in?

My point is that 2^k+1 + 2^k+1 doesn't equal 4^k+1.

Replace k with 1:

2^(1+1) + 2^(1+1) = 4^(1+1)

2^2 + 2^2 = 4^2

4 + 4 = 16

8 = 16.

It doesn't work.

However 2^k+1 + 2^k+1 does equal 2 * 2^k+1, which is where you can factor out the 2. For the same reasons as above, you cannot do 2 * 2^k+1 and get 4^k+1.

(September 26, 2018 at 9:56 pm)Tiberius Wrote:

(September 26, 2018 at 9:42 pm)Aliza Wrote: Purplemath factored out the 2 at an earlier stage.
    = [2^k+1 – 2] + 2^k+1
I added the 2^k+1 to the other 2^k+1 to give me 4^k+1, and then I factored out the 2 from the remaining two terms.

It is a proof. I'm just trying to get from point A to point B, but I can't for the life of me figure out what exponent rule allows for that. Why if I can factor out a 2 and then redistribute it does it not just go back into the equation without affecting the exponent? I understand that (x^2)(x^2) would be x^(2+2) which would be x^4, but why in the case of my example question, did the exponent not change when I factored out that two as such that it would go up by one when I put the two back in?

My point is that 2^k+1 + 2^k+1 doesn't equal 4^k+1.

Replace k with 1:

2^(1+1) + 2^(1+1) = 4^(1+1)

2^2 + 2^2 = 4^2

4 + 4 = 16

8 = 16.

It doesn't work.

However 2^k+1 + 2^k+1 does equal 2 * 2^k+1, which is where you can factor out the 2. For the same reasons as above, you cannot do 2 * 2^k+1 and get 4^k+1.

Hmmmm. A few days ago I was redirected to do it this way, but what you're saying is also making sense. I remember thinking a few days ago, "what the fuck are you talking about?!" But maybe I misunderstood what I was being told and trying to jam this new exponent understanding into my previous understanding shorted out my brain.

Because I sure as hell wasn't going to say to the guy, "Wait, can you explain that to me again. That doesn't make sense."

I've been looking at this way too long... and now I'm all twisted around.