I have a simple problem which gives two different solutions in two different calculators.

Question

I hope that this kind of questions doesn't break the site's rules.

I have this simple problem which gives two different solutions in two different calculators (Wolfram Alpha and Symbolab).

What am I doing wrong in the calculation of $$ [(-7)^6]^\frac{1}{2} \:\:? $$ In one of the solutions, the solution is positive, and negative in the other. I think that the problem might arise when trying to solve a power of a power with a negative base. Is there a restriction to this kind of problems?

Here are the two links to the solutions:

Wolfram Alpha

Symbolab

Thanks in advance for your help!

Kind regards!

Answer 1

As you are taking a square root, either answer will work.

$(-a) \cdot (-a) = a^2$

$a \cdot a = a^2$

Wolfram Alpha seems to be giving you the principle root, while the other calculator seems to be giving you the secondary root.

Answer 2

(-7)^6 should always return a positive number. If it isn't, you're forgetting the negative sign inside the operand. Square root of a positive number is positive, so the answer should be positive.

Answer 3

Symbolab used the rule $(a^m)^n=a^{m×n}$, which does not always work when $a$ is negative and $m$ and $n$ are not integers; see this answer.

Answer 4

Both answers are correct, in that $343^2$ and $(-343)^2$ both equal $(-7)^6$.

There is no the square root of anything (except $0$) because $\sqrt x$ or $x^{\frac12}$ is a 2-valued function.

That said, you may feel that thinking “the square root of a sixth power is the same as the cube” is over-clever. But on the other hand you may feel that taking it as the cube is rescuing you from the loss of sign information caused by raising to the sixth power in the first place.