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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?

vonbrand
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Bill
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    Are you sure you're putting parentheses around the $-64$? If you enter it as $\displaystyle -64^{\frac{2}{3}}$ the calculator might interpret it as $\displaystyle -\left(64^{\frac{2}{3}}\right) = -16$. – 2012ssohn Feb 27 '14 at 02:33
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    Not just "might": it will. (Exponents and roots are done first, then the sign is appended.) – colormegone Feb 27 '14 at 02:37
  • The calculator in Windows gives 16 as the result (when calculating $(-64)^{2/3}$). – Martin Argerami Feb 27 '14 at 02:38
  • I think if you enter $-64$ and then something like Ans^(2/3) it would actually interpret it as $\displaystyle (-64)^{\frac{2}{3}}$. I know it does on a TI-83 or 84. So, it might, depending on how he entered it. – 2012ssohn Feb 27 '14 at 02:41
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    I just pulled out four calculators of mine and tried it. Three gave -16 and the fourth one choked. I agree with 2012ssohn's interpretation: if you enter the problem directly, the "hierarchy of operations" will more likely than not give you -16, but using the "Ans" feature essentially "wraps parentheses" around the " -64" and then you can perform the powers-calculation correctly. – colormegone Feb 27 '14 at 02:45

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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 $$ a^{1/3} = \sqrt[3]{a}. $$

Thomas
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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.

kitkat
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  • This just illustrates a maxim going back to early electronic-computing days: you should know what the machine does in performing the calculation before asking it to do it for you... – colormegone Feb 27 '14 at 02:48