If square root is the inverse function of $5^2$ what is the inverse function of $5^1$

Question

I am not great at maths or anything, but just had a general question:

If square root is the opposite of $5^2$, what is the opposite of $5^1$, $5^3$, $5^4$?

Is there an opposite? How would I work it out?

I may be having a brain freeze but thank you for your help.

Answer 1

The inverse of the map $x\mapsto x^d$ is $x\mapsto x^{1/d}$ for $d\in\mathbf{R}_{+}^{\star}$. By opposite I guess you mean the inverse function, right ?

Answer 2

I think you mean inverse rather than opposite. The inverse of square is squareroot which could be expressed as:$$f(x)=x^2$$$$\therefore f^{-1}(x)=\sqrt{x}=x^{\frac{1}{2}}$$In a similar vein the inverse of cubing a number would be expressed as:$$f(x)=x^3$$$$\therefore f^{-1}(x)=\sqrt[3]{x}=x^{\frac{1}{3}}$$And in general:$$f(x)=x^n$$$$\therefore f^{-1}(x)=x^{\frac{1}{n}}$$ NOTE: When $n=1$ you get a function that is its own inverse, i.e.:$$f(x)=x$$$$\therefore f^{-1}(x)=x$$This is called the identity function.