+0

# Write the three cube roots of -27 in rectangular coordinates. Give exact answers.

0
1329
1

Write the three cube roots of -27 in rectangular coordinates. Give exact answers.

Guest May 28, 2014

#1
+1792
+14

$$\sqrt[3]{-27}=\sqrt[3]{27}\sqrt[3]{-1}=3\sqrt[3]{-1}$$

$$\sqrt[3]{-1}=\sqrt[3]{e^{\imath \pi}}=\sqrt[3]{e^{\imath (\pi +2k \pi)}}~~k\in \mathbb{Z}$$

and we pick values corresponding to k=0,1,2

$$\sqrt[3]{-1}=e^{\imath \pi/3}$$

$$\sqrt[3]{-1}=e^{\frac{\imath(\pi + 2\pi)}{3}}=e^{\imath\pi}=-1$$

$$\sqrt[3]{-1}=e^{\frac{\imath(\pi + 4\pi)}{3}}=e^{\imath 5\pi/3}=e^{-\imath \pi/3}$$

so

$$\sqrt[3]{-27}=-3, ~3e^{\imath \pi/3}, ~3e^{-\imath \pi/3}$$

Oh I see it wants the answers in rectangular form

To do this, convert the complex exponential into rectangular form as follows

$$e^{\imath x}=\cos(x)+\imath \sin(x)$$

$$-3 = -3 \\$$

$$3e^{\imath \pi/3} = 3(1/2 + \imath \sqrt{3}/2)=3/2+\imath 3\sqrt{3}/3$$

$$3e^{-\imath \pi/3} = 3(1/2 - \imath \sqrt{3}/2)=3/2-\imath 3\sqrt{3}/3$$

Rom  May 28, 2014
Sort:

#1
+1792
+14

$$\sqrt[3]{-27}=\sqrt[3]{27}\sqrt[3]{-1}=3\sqrt[3]{-1}$$

$$\sqrt[3]{-1}=\sqrt[3]{e^{\imath \pi}}=\sqrt[3]{e^{\imath (\pi +2k \pi)}}~~k\in \mathbb{Z}$$

and we pick values corresponding to k=0,1,2

$$\sqrt[3]{-1}=e^{\imath \pi/3}$$

$$\sqrt[3]{-1}=e^{\frac{\imath(\pi + 2\pi)}{3}}=e^{\imath\pi}=-1$$

$$\sqrt[3]{-1}=e^{\frac{\imath(\pi + 4\pi)}{3}}=e^{\imath 5\pi/3}=e^{-\imath \pi/3}$$

so

$$\sqrt[3]{-27}=-3, ~3e^{\imath \pi/3}, ~3e^{-\imath \pi/3}$$

Oh I see it wants the answers in rectangular form

To do this, convert the complex exponential into rectangular form as follows

$$e^{\imath x}=\cos(x)+\imath \sin(x)$$

$$-3 = -3 \\$$

$$3e^{\imath \pi/3} = 3(1/2 + \imath \sqrt{3}/2)=3/2+\imath 3\sqrt{3}/3$$

$$3e^{-\imath \pi/3} = 3(1/2 - \imath \sqrt{3}/2)=3/2-\imath 3\sqrt{3}/3$$

Rom  May 28, 2014

### 36 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