Zeroth root

√16 = 4. This means that 42 (4*4) = 16. 3√27 = 3. This means that 33 (3*3*3) = 27. Now what's 0√38? If you can't think of an answer, that's okay. Taking the zeroth root of a number isn't really touched on in math, so there's no real definition.

Roots are defined as a number r so that rn = x; this means that x$1⁄n$ = r. Your next thought might be to assign 0 to n; this results in the equation x$1⁄0$ = r. You may see the problem here: $1⁄0$ is undefined. You could stop here: since 0√x is equivalent to x$1⁄0$, which results in a division by 0, 0√x is undefined. However, I'll keep going and attempt to define $1⁄0$.

Take the expression $1⁄x$. As x becomes infinitely large, $1⁄x$ approaches 0: $1⁄10$ = 0.1, $1⁄100$ = 0.01, $1⁄1,000$ = 0.001, etc. As x approaches 0, the opposite happens: $1⁄0.1$ = 10, $1⁄0.01$ = 100, $1⁄0.0001$ = 1,000, etc. This is the concept of a limit: $1⁄x$→∞ (infinity) as x→0. For our purposes, we'll say that $1⁄0$ = ∞.

Since we've defined $1⁄0$ as infinity, 0√x should be infinity, right? Not quite. Although x∞ = ∞ if x > 1 (for example, 21 = 2, 210 = 1,024, 220 = 1,048,576, 230 = 1,073,741,824, etc.), it actually equals 0 if x < 1 (for example, 0.51 = 0.5, 0.510 = 0.0009765625, 0.520 = 0.000000953674316, etc.). And obviously, it'll remain at 1 if x = 1 (no matter how much you multiply 1 by 1, the result will still be 1). Does that mean 0√x can't be defined, since the result depends on the value of x? Actually, no: in math, we're allowed to assign multiple different values to a function depending on the input value (this is called a piecewise definition and is normally denoted with a left-facing curly bracket, {). So, here's a possible piecewise definition for 0√x:

0√x =
 * ∞, if x > 1
 * 1, if x = 1
 * 0, if -1 < x < 1

You might've noticed that the definition doesn't include negative numbers less than -1. This is for a good reason: if you raise a negative number to a power, the sign of the result depends on if the power is even or odd (for example, (-1)2 = 1, but (-1)3 = -1). While this arguably isn't an issue with negative numbers greater than -1 ((-0.5)∞ equals 0, which doesn't have a sign anyway), it means that the zeroth root cannot be defined for numbers such as -2 (should (-5)∞ be positive or negative ∞?).