Power of power math question?

Question

Is
(x power y) power z = x power yz
only true if
x >= 0
?

Please explain why.

Answer 1

The calculation of power is done by Taylor series and the formula x power y = exp(y * ln(x))
ln(x) is only defined for x>0. So we cannot take the power of a negative number unless the power is a whole number.
If that number is a whole number, this is if y is a whole number, then we can take the power but if z is not a whole number then x power y has to be positive which is only the case if y is an even whole number (when x < 0).

Answer 2

The reason for requiring x to be non negative is not the rule you are referring to, but rather is a consequence of the definition of raising to a fractional power. If the power is a fraction with an even denominator, then you would have to take an even root of x, which does not give you a real number if x is negative.

eg, "x power y" is undefined if x = -4 and y = 1/2.

Answer 3

If y and z are integers, then (x^y)^z = x^(y*z) for any real value of x (actually, it holds for imaginaries as well, but we won't go there). If y and z are allowed to be any real numbers, then x must be non-negative if either y or z is irrational. If y and z are allowed to be any rationals, things get interesting.

Certainly if x>=0 we don't have any problems. So let's assume x=-1 (a nice, simple example). Then, depending on whether we want to delve into imaginary numbers, (x^(1/2))^2 is either undefined (we get sqrt(-1) and give up), or -1 (we get sqrt(-1) and square that to get back to -1). Obviously the second choice allows us to simplify the exponent using your rule, but if we would prefer to avoid any reference at all to imaginary numbers, then the rule does not hold.

So, for a more concise answer to your question, I would say (on the real numbers), yes. The rule only works consistently for x>=0.

Answer 4

The rule is that powers of powers are multiplied. Look:

(2^2)^3 = 2^6

if you string this out you have: (2^2)(2^2)(2^2) or 2x2x2x2x2x2

Don't get it confused with adding powers when multiplying:
2^3 x 2^2 = 2^(2+3) = 2^5.

hope this helps!