What is power of a product?

Answer 1

The simple answer is that it is the product of the powers:

(ab)^n=a^n*b^n.

However, there is one caveat: a and b must be positive.

Consider:

((-1)*(-1))^0.5

=(1)^0.5

=1

OR

((-1)*(-1))^0.5

=(-1)^0.5 * (-1)^0.5 (by the rule stated above)

=i * i (since i, the imaginary unit, is defined so that i=sqrt(-1))

=-1 (since if i=sqrt(-1), i^2=-1).

This expression cannot be both 1 and -1 at once (remember that we are taking the primary root here)... clearly there is a problem.

This problem comes into play when we take non-integer exponents of non-real-positive numbers.

A modification that holds in general is that

|(a*b)^n|=|a^n * b^n|, where |f(x)| is the absolute value of f(x).

As you can see, this works for the above example, as |-1|=|1| (=1).

Answer 2

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