Having a little trouble with this one, wolfram returned ago an answer however looking because that a more thorough answer.
You are watching: Cube root of x^2
it is troublesome. The most basic for girlfriend is come only enable cubed root of x, squared for x>0. If you feeling adventurous, include zero as feasible argument because that x, but stay far from an adverse numbers.
the exponential rule (xa)b = xab = (xb)a is no valid no longer for non-positive x. Granted, you will find ways to make it work-related still even with an unfavorable x and also there has been a very good recent write-up by u/Midtek in the Math ar of r/askscience replying to a connected question, that explains the conditions as to why and also when.
· 7 yr. Ago
What are you talking about? There's no domain concerns for x2/3
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· 7 yr. Back · edited 7 yr. Ago
It is true that identities prefer (zw)a = zawa perform not necessarily hold for all complex numbers z and w if a is no an integer. I assume you room referencing my short article here which discusses the identity for a = 1/2. It transforms out that for the square root, the identification holds for every non-negative real numbers, as lengthy as the major branch is chosen. The identity does not host for negative real numbers, unless the various other branch is chosen.
For the cube root, us run right into the same problem. The main branch that the cube source is identified so that if z = |z|eiθ with θ in the expression <0, 2π), then
z1/3 = |z|1/3eiθ/3
(The cube source of |z| is simply the usual cube root on non-negative actual numbers.) Then we may consider the identity
(zw)1/3 = z1/3w1/3
If z = w = -1, climate the left-hand next is 1 but the right-hand next is ei 2π/3 = -1/2 + i√(3)/2. So the identification does not hold for all facility numbers, even for the cube root. If you input (-1)1/3 into Wolfram Alpha you acquire the major cube root, i beg your pardon is ei π/3 = +1/2+ i√(3)/2.
However.... The is because that the cube source on the complex numbers, i m sorry is constant with the cube source on non-negative real numbers. Over there is a different definition of the cube root for an unfavorable real numbers, which is regular with the cube root because that non-negative actual numbers but not consistent with the cube source on complicated numbers. So frequently you will certainly see human being write
(-1)1/3 = -1
Note the this equation is emphatically not continual with the cube source for complex numbers.
The cube root function on genuine numbers is defined so the x1/3 gives the unique real number y such that y3 = x. It transforms out then the under this definition, (-1)1/3 = -1. This meaning is regularly the only definition taught come high school students or to university students before complex analysis. Wolfram Alpha go not usage this alternate definition.
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Consequently, the function f(x) = x2/3 is characterized to it is in the square the the duty x1/3. But, again, keep in mind that this an interpretation is not regular with the definition of z2/3 because that general facility z.