(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
