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Scrolling through questions I see that someone has claimed that the two are not the same. Why is this so? And how would this effect evaluating the derivative of $\frac{3}{4}x\sqrt[3]{x}$ at $x=-27$ vs. the derivative of $\frac{3}{4}x^{4/3}$ at $x=-27$.

What is the Domain of $f(x)=x^{\frac{1}{x}}$

Community
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Ahmed S. Attaalla
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1 Answers1

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The radix ,written as $\sqrt[3]{-27}$, refers specifically to the normal form of $(-27)^{1/3}$. In the case of negatives, the normal form also includes a negative number.

The form $(-27)^{1/3}$ refers to any solution of the form $a^3=-27$, but when one is dealing with several values derived from $a^3=-27$, one should always use the same root. In practice, using the radix-form is the normalized first root of the equation.

Arbuja
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wendy.krieger
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  • So the only solution to $\sqrt[3]{-27}$ is $-3$? – Arbuja Dec 14 '15 at 12:35
  • Yes. They do this, because if one needs to access the other cube roots, you multiply it by a constant. So while, eg [math]{4}^{1/2}=2, -2[/math], you write these as [math]\sqrt{4}, -sqrt{4}[/math]. Then you can rely on [math]\sqrt{4}[/math] being +2 – wendy.krieger Dec 14 '15 at 13:26
  • Then why for text books is $x^{1/2}$ a function. This CAS standards versus general mathematical standards. Perhaps you modify the CAS version of $x^{1/2}$ into $x=y^2$. – Arbuja Dec 15 '15 at 13:51