I get easily confused while evaluating exponents: when should I add them and multiply them?
- We add exponents when multiplying two numbers are of the same base: i.e., $ (2^2) . (2^3) = 2^5 $ - When do we multiply exponents?
- For example: $ 2^{4^2} $ and $ (2^4)^2 $are different things. I usually resort to using a calculator or webapp to make sure I'm doing this correctly. Why can we write $ (2^4)^2 = 2^ 8$ but not $ 2^{4^2} = 2^8 $?
- When can I "group" exponents on numbers with different bases? For eg: can I write: $ (2^2) . (4^2) = (8^2) $? How about: $ \sqrt{4} . \sqrt{8} = \sqrt{32} $?
What are other operations on exponents that I need to remember without resorting to a calculator? The first one comes naturally to me, since we can expand it out as $ 2 . 2 . 2 . 2 . 2 $. The others always evade my intuition.