Is a variable with no power assumed to be to the power of one?

Answer 1

Yes. If we don't see an exponent, the exponent is understood to be one. Just like if we don't see a sign, the sign is assumed to be positive. This in no way constricts or limits the variable. The variable can still be anything at all. But assumptions like this are part of mathematics. How tedious would it be to have to write "x1" for each x? In any case, x1 = x so it is almost pointless.

Yes, because if the variable (hypothetically speaking) is 1 (and it can be any number in the world, but 1 is an example), and the exponent (power) is 1, there is only one number to multiply, therefore, the number can only be by itself. If it is squared (to the second power) then you multiply the variable against itself (in this case, 1 X 1), but if the variable doesn't have an exponent, it is assumed to be to the power of one. Take the number 2, for example. If 2 is the variable, x, and x is alone, and you multiply how big the number is of the exponent, and because there is none, it is to the first power.

X squared looks like this: x X x

X (a.k.a. x to the first power) looks like this: x

If there is no exponent, you only multiply what is there, which there is only one variable, so it is assumed to be to the first power. If it were squared, or to the second power as demonstrated above, you would multiply x by itself. If you count the variables in the equation, x squared has two x's, while x alone has only one x in the equation. Because there is only one x, and the exponent represents how many of that number (in this case, x) you multiply by itself, x is assumed to be to the first power, or to the power of one.