How to simplify $2^{(3+(4+(5+(6+(7+(8))))))}$?

Question

Thanks to the rules of parenthesis in exponents, $2^{(3+(4+(5+(6+(7+(8))))))}$ has a huge number of parenthesis.

How do you simplify this so you wouldn't have to write $))))))$ at the end of a math problem?

Answer 1

$2^{(3+(4+(5+(6+(7+(8))))))} = 2^{(3+4+5+6+7+8)} = 2^{33}$. The only set of parantheses needed is the outermost one, to imply that you do the addition before you do the exponentiation.

Answer 2

$3+(4+(5+(6+(7+(8))))) = 3+4+5+6+7+8 = 33$

Therefore, $2^{(3+(4+(5+(6+(7+(8))))))} = 2^{33}$.

Answer 3

This would be identical to simply simplifying the exponent.

So what would an equivalent expression be for the exponent, i.e. what does

$$(3+(4+(5+(6+(7+(8)))))))$$

equal? (Or the sum with the numbers up to $6$, you have two different expressions in the question.) Obviously it would just be the sum of these numbers, right? Thus, you can just substitute the sum into the exponent. (This applies to either one.)

Answer 4

Addition is associative so in your case, we have:

$$2^{(3+(4+(5+(6))))}=2^{(3+4+5+6)}$$

No need for an inordinate number of parentheses here.

Answer 5

To combat this kind of problem when typesetting documents, we often either just evaluate the exponent (like Mike suggested), or use dots, like $2(3+(4+(5+\cdots+20(21))))$. Alternatively, I've seen people use $)\cdots)$, but that's exceedingly rare. If working in a context when you have commutativity and associativity though, the first option is probably the way to go.