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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

Best Answer 

 #1
avatar+20009 
+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
 #1
avatar+20009 
+13
Best Answer

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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