+0  
 
0
284
1
avatar+38 

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+18831 
+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
Sort: 

1+0 Answers

 #1
avatar+18831 
+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

9 Online Users

We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details