In algebra, why are all odd powers reversible?

Question

We're working on the use of backtracking to solve a problem. Also, using reversing or undo a series of steps to solve equations. The question states," Make tables for the outputs x^4, x^5, x^6 and x^7. Which powers produce reversible operations?." The answer is all odd powers are reversible. How can I show this by example?

Answer 1

Strictly speaking "NO" power operation (raising to an exponent) is reversible, because the inverse operation is ALWAYS multi-valued, unless we restrict the domain. In the case of even powers, for example:

(+3) ^2 = 9
(- 3) ^2 = 9

Therefore the reverse operation thus has two possible solutions:

Sqrt(9) = +3
Sqrt(9) =  -3

Also, it is "NOT" true that odd powers are reversible, if we consider complex numbers:

(+3                        ) ^3  = 27
(-1.5 + j 2.59808) ^3  = 27
(-1.5 -  j 2.59808) ^3  = 27

The cube root of three (above) therefore has a total of three solutions, one "real"  and two complex conjugates. In the general case, the inverse operation of:

y = (x)^n        --->   x = nth root of (y)

has "n" solution. Odd roots(y) have a single real solution and (n-1)/2 pairs of complex conjugate solutions. Even roots of (y) have two real solutions (+/-) and (n-2)/2 pairs of complex conjugate solutions, assuming that "y" is positive.

In the general case, the "nth root" of any real or complex power can be plotted as a set of equally spaced vectors in the complex plane, each with a magnitude of:

   (|y|  / n)

separated by angles of:

  (2 pi / n)   or    (360 degrees /n)

 

In summary:

    Even powers are reversible if we restrict our domain to positive numbers.
    Even powers are NOT reversible in the INTEGER or larger domains,
    because the roots are multi valued.

    Odd powers are reversible in the REAL domain,
    but NOT in the COMPLEX or larger domains.

Hope this helps,

MJS

Answer 2

Correction (typo) in the magnitude calculation:

| n-th root of x | = n-th root (|x|)

i.e.

a)   The "n"-th root of (x) has "n" distinct solutions.

b)  All solutions have the same magnitude = "n"-th root of the original magnitude,

c) The roots are equally spaced (equal angles) on a circle in the complex plane.