Exponentiation is a correspondence between addition and multiplication. Think of a number line, with $0$ in the "middle", and tick marks at each integer. Moving a certain distance to the right corresponds to adding a positive number, and adding the same number moves you the same distance, no matter where you are on the line.
I was wondering why addition has one inverse (subtraction), multiplication has one inverse (division), but exponentiation has two (radication and logarithm). After a bit of thinking, I thought it m...
You know, like addition is the inverse operation of subtraction, vice versa, multiplication is the inverse of division, vice versa , square is the inverse of square root, vice versa. What's the in...
Take the following: (2)^3 = 8 I understand that this is 2 * 2 * 2 = 8 My question is how do I reverse engineer this if I do not know the power like this: (2)^x = 8 What is the value of x? x could
When rigorously laying out foundations of all of analysis, the usual method is to use the definition $$ a^b = \exp (b \log (a)) $$ While you could try and define exponentiation directly, it is somewhat awkward. Since you have to define $\exp$ and $\log$ anyways, it's most convenient to take advantage of that work, and the fact this identity is so simple. This definition also has an advantage ...
We use multiplication for repeated addition, and in turn use exponents for repeated multiplication. What topic comes after this, for repeated exponentials? Is there something my teachers are hiding...
Rather than trying to press complex exponentiation into the mold of repeated multiplication, see complex exponentiation -- or more fundamentally, the function $\exp$ -- as its own thing, that in special cases can be interpreted as repeated multiplication, thanks to the functional relation $\exp (a)\exp (b) = \exp (a + b)$.
We can define exponentiation as repeated multiplication and solve the problem of "what is the nth root of something" by defining it as "it's whatever thing that, when you multiply it by itself n times, gives you that something." That's what your question is essentially about.
6 They are closed under exponentiation, the contradiction you gave isn't a real contradiction. When you say that there can be "up to an infinite amount" of zeroes, that just means that there is no upper bound on the number of trailing zeroes. However any specific number you choose from that set will have some finite number of trailing zeroes.
