SOLUTION: This one is really bugging me: I understand that 2(2^n-1) (that is, two times two to the n-1 power) is equal to 2^n since the rule is "add the exponents if the bases are equal")

Question 397013: This one is really bugging me:
I understand that 2(2^n-1) (that is, two times two to the n-1 power) is equal to 2^n since the rule is "add the exponents if the bases are equal"). But why can't I compute it this way: 2(2^n-1) = 4^n-1 which could be rewritten as (2^2)^n-1 which, following the rule "multiply exponents when you have something to a power taken to another power" would then distribute as 2^2(n-1) = 2^2n-2. But this is a completely different answer!
This doesn't make any since so surely I am making an obvious mistake. But given, for example 2(x^n+1) don't get 2x^n+1???
Thank you!
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
This is what's going on (skip this part if you know this already)

Start with the given expression.

Rewrite as

Now we're going to use the identity

Multiply (by adding exponents) using the identity given above

Add

So

----------------------------------------------------

Now your question is why isn't equal to . The simple reason why not is because you can't simply multiply the bases. Remember that is really

For example, we know that . So

But if we multiply the bases, then which is NOT true.

Also, you CANNOT rewrite into because the two are completely different. If you're skeptical about that, plug various values of 'n' into each expression and you'll see different results.

So unfortunately, your line of reasoning was based on a false assumption which lead to that contradiction.

If you need more help, email me at jim_thompson5910@hotmail.com

Also, feel free to check out my tutoring website

Jim
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