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# fun with square

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Why is (-5)^3=-125 (-5*-5*-5)=-125,

and 3-th square of (-125) is not defined in real numbers, only in complex numbers? ((-125)^1/3))??? any idea?

blaster0  May 28, 2014

#1
+19062
+13

3-th square of (-125):

$$\\\sqrt[3]{-125}=\sqrt[3]{125}\times\sqrt[3]{-1}=(5)* \left\{ e^{i\frac{1}{3}(\pi+0*\pi)}, e^{i\frac{1}{3}(\pi+2*\pi)}, e^{i\frac{1}{3}(\pi+4*\pi)} \right\}\\ \sqrt[3]{-125}=(5)* \left\{ e^{i\frac{1}{3}\pi}, e^{i\frac{3}{3}\pi}, e^{i\frac{5}{3}\pi} \right\}\\ \sqrt[3]{-125}=(5)* \left\{ e^{i\frac{1}{3}\pi}, e^{i\pi}, e^{i\frac{5}{3}\pi} \right\}\\ \boxed{e^{i\pi}=-1}\\ \sqrt[3]{-125}=(5)* \left\{ e^{i\frac{1}{3}\pi}, -1, e^{i\frac{5}{3}\pi} \right\}\\ \\ \sqrt[3]{-125}=(5)* e^{i\frac{1}{3}\pi} \quad \text{complex number} \\\\ \sqrt[3]{-125}=(5)*(-1)=-5\quad \text{real number }\\\\ \sqrt[3]{-125}=(5)* e^{i\frac{5}{3}\pi} \quad \text{complex number} \\$$

heureka  May 29, 2014
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#1
+19062
+13
$$\\\sqrt[3]{-125}=\sqrt[3]{125}\times\sqrt[3]{-1}=(5)* \left\{ e^{i\frac{1}{3}(\pi+0*\pi)}, e^{i\frac{1}{3}(\pi+2*\pi)}, e^{i\frac{1}{3}(\pi+4*\pi)} \right\}\\ \sqrt[3]{-125}=(5)* \left\{ e^{i\frac{1}{3}\pi}, e^{i\frac{3}{3}\pi}, e^{i\frac{5}{3}\pi} \right\}\\ \sqrt[3]{-125}=(5)* \left\{ e^{i\frac{1}{3}\pi}, e^{i\pi}, e^{i\frac{5}{3}\pi} \right\}\\ \boxed{e^{i\pi}=-1}\\ \sqrt[3]{-125}=(5)* \left\{ e^{i\frac{1}{3}\pi}, -1, e^{i\frac{5}{3}\pi} \right\}\\ \\ \sqrt[3]{-125}=(5)* e^{i\frac{1}{3}\pi} \quad \text{complex number} \\\\ \sqrt[3]{-125}=(5)*(-1)=-5\quad \text{real number }\\\\ \sqrt[3]{-125}=(5)* e^{i\frac{5}{3}\pi} \quad \text{complex number} \\$$