why is $(-64)^{2/3} =-16$ and not $16$?

Question

It appears that taking the cube root of a negative number will yield a negative number, which when squared, will yield a positive number. But all the calculators and books I have seen show this particular problem yields a negative number. Any help here?

Answer 1

Indeed $$ (-64)^{2/3} = 16. $$ This can be seen two ways: $$\begin{align} (-64)^{2/3} &= ((-64)^{1/3})^2 = (-4)^2 = 16 \\ (-64)^{2/3} &= ((-64)^2)^{1/3} =(4096)^{1/3} = 16. \end{align} $$ Here remember that $$ a^{1/3} = \sqrt[3]{a}. $$

Answer 2

Well let's take this step by step though you do seem to have an understanding of it.

The cube root of -64 is -4. So yes It does yield a negative number. Then the square of that is 16, so like the previous person said perhaps you have not entered it into the calculator properly.

Ah technology, a blessing but yet a curse.