How is n to the power of less than one possible?

[Guest]

Hello! I hope I'm posting in the right place.
So in math class our teacher once said "n to the power of half" [tex]n^{\frac{1}{2}}[/tex]
I know that it equals the square root of n but I don't know why, and I don't seem to understand how, for example, [tex]2^{\frac{1}{2}}[/tex] works. All that means to me is multiplying 2 by itself 0.5 times. That seems impossible and not logical! I also ask the same question for other less-than-one-powers like [tex]5^{\frac{1}{3}}[/tex] for instance.
If anyone can explain this to me I will be really grateful, and thanks.
PS: if my question doesn't match where i posted it, plz tell me so I can delete and post it where it belongs.

[Baltuilhe]

Good afternoon!

In this specific case (decimal numbers, not irrational), you can use this idea:
[tex]\overbrace{x\cdot x\ldots x\cdot x}^{\text{n values}}=x^n\\\\x=y^{1/n}\\\\\overbrace{y^{1/n}\cdot y^{1/n}\ldots y^{1/n}\cdot y^{1/n}}^{\text{n values}}=\left(y^{1/n}\right)^n\\\\y^{1/n}\cdot y^{1/n}\ldots y^{1/n}\cdot y^{1/n}=y^{(1/n)\cdot n}=y[/tex]

Helped?

[HallsofIvy]

The idea that "[tex]x^n[/tex]" means "multiply x by itself n times" applies only if x is a "counting number" (positive integer). In order to talk about "[tex]x^n[/tex]", for n not a positive integer, we have to define what that means!

We can, of course, define things however we want but it would be nice to define them so that they have nice properties. One nice property that the previous definition has is that [tex](x^n)(x^m)= x^{m+ n}[/tex]. That is, if I am multiplying n copies of x and another m copies of x then I am multiplying m+ n copies of x together, Another is that [tex](x^n)^m= x^{mn}[/tex]. To see that, imagine the first n copies of x written on one line, the second n copies written on a second, line, etc. until you have m lines, each with n copies of x. We can think of that as a "rectangle" of "x"s, n across by m high: a total of mn "x"s.

But what could we possibly mean by [tex]x^0[/tex]? We can't "multiply x by itself 0 times"! No, we can't so we need a separate definition. We would like, very much, for the nice property "[tex](x^n)(x^m)= x^{m+ n}[/tex]" to still be true even if one of the powers is 0. That is, we want [tex](x^n)(x^0)= x^{n+ 0}[/tex]. But n+ 0= n. So we want [tex](x^n)(x^0)= x^n[/tex]. In order for that to be true we must define [tex]x^0= 1[/tex] for all x.

What about x to a negative power? Again, we want [tex](x^n)(x^m)= x^{m+ n}[/tex] even when one or both of the powers is negative. In particular, we want [tex](x^n)(x^{-n})= x^{n+(-n)}= x^0= 1[/tex]. That means that we want [tex]x^{-n}= \frac{1}{x^n}[/tex] so that is how we define x to negative power. (Of course, that means we must restrict x to be non-zero.)

What about x to a general rational number power (including your 1/2 power)? Here we want [tex](x^n)^m= x^{mn}[/tex] to remain true. That is, we want [tex](x^{1/n})^n= x^{n/n}= x[/tex]. But that clearly requires that [tex]x^{1/n}= \sqrt[n]{x}[/tex] (and so [tex]x^{1/2}= \sqrt{x}[/tex]). More generally, any rational number is of the form [tex]\frac{m}{n}[/tex] with m and n integers. We define [tex]x^{\frac{m}{n}}= \sqrt[n]{x^m}[/tex]. (Of course, that requires that x be positive. In general, we define the function [tex]a^x[/tex] only for a positive.)

To go to all real number powers, we have to include the "irrational numbers" that cannot be written as fractions. In fact, the real numbers cannot be defined "algebraically"! They have to be defined "analytically", using some sort of limiting process. The simplest is just to assert that for every real number, x, there exist a sequence of rational numbers, [tex]\{x_m\}[/tex], that converges to x. For example, [tex]\pi[/tex] is not a rational number so cannot be written as fraction (22/7 is only an approximate value). But any real number can be written as a decimal: [tex]\pi= 3.141592...[/tex] where the "..." means that it just keeps on going. [tex]\pi[/tex] is the "limit" of the infinite sequence of rational numbers, 3, 3.1, 3.14, 3.141, 3.1415, 3.14159, 3.141592, ... . And then we define [tex]x^{\pi}[/tex] to be the limit of the infinite sequence [tex]x^3[/tex], [tex]x^{3.1}[/tex], [tex]x^{3.14}[/tex], [tex]x^{3.141}[/tex], [tex]x^{3.1415}[/tex], [tex]x^{3.14159}[/tex], [tex]x^{3.141592}[/tex], ...