#13**+10 **

Look at the step where fiora sets -1 + √3i to ^{3}√(-1 + √3i)^{3}

In the next step (-1 + √3i)^{3 }is expanded, correctly, to get 8, but then the wrong cube root of 8 is taken. Fiora says the cube root of 8 is 2. If this were the right thing to do here then it would mean that -1 + √3i equals 2, which would be a little unusual to say the least!

The fact is that, in the domain of complex numbers, there are three cube roots of 8, namely 2, -1 - √3i and -1 + √3i. The only valid root here is the last one. It isn't valid to jump from -1 + √3i to (-1 + √3i)^{3} and then jump back to a different root from the one you came from!

It's like saying -2 squared is 4, the square root of 4 is 2, so we can replace -2 with 2.

.

Alan
Apr 21, 2015

#4**+5 **

Hi Fiora, welcome to the forum.

Did I talk to you before and suggest that you join and that you take that icon?

It looks great :))

-------------------------------------------------

Edited: this was wrong

Melody
Apr 21, 2015

#6**+10 **

Sorry Badinage and fiora, I think you are right - I was suffering from brain freeze

I have not looked from there on Badinage, is the rest of Fiora's work correct?

$$\\(-1+\sqrt3i)^3\\\\

=(\sqrt3i-1)^3\\\\

=[(\sqrt3i)^3]+[3(\sqrt3i)^2(-1)]+[3(\sqrt3i)(-1)^2]+[-1^3]\\\\

=[3\sqrt3*-i]+[-3(3*-1)]+[3(\sqrt3i)]+[-1]\\\\

=[-3\sqrt3i]+[9]+[3\sqrt3i]+[-1]\\\\

=+[9]+[-1]\\\\

=+8$$

Melody
Apr 21, 2015

#8**+10 **

ok then, what about

$$\\(1+\sqrt3i)^3\\\\

=(\sqrt3i+1)^3\\\\

=[(\sqrt3i)^3]+[3*(\sqrt3i)^2]+[3*(\sqrt3i)]+1\\\\

=(3\sqrt3*-i)+(3*3*-1)+(3*\sqrt3i)+1\\\\

=(-3\sqrt3i)+(-9)+(3\sqrt3i)+1\\\\

=-8$$

SO

$$\\\sqrt[3]{ \frac{14*\sqrt[3]{(1+3i)^3} }{2}}\\\\

=\sqrt[3]{ \frac{14*\sqrt[3]{-8} }{2}}\\\\

=\sqrt[3]{ \frac{14*\sqrt[3]{8}*\sqrt[3]{-1} }{2}}\\\\

=\sqrt[3]{ \frac{14*2*\sqrt[3]{-1} }{2}}\\\\

=\sqrt[3]{ 14\sqrt[3]{-1} } \\\\$$

I have yet to consider $$\sqrt[3]{-1}$$

Does that look right or wrong to you Badinage?

Melody
Apr 21, 2015

#9**+5 **

I had a quick look at Wolfram|Alpha and I cannot tell whether it agrees with mine or not since mine is not finished :)

Melody
Apr 21, 2015

#10**+10 **

Now I have to consider

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

$The first obvious answer is $ -1+0i $ \;\;since (-1)^3=-1\\

$Now, there will be 3 roots$\\

$each will be $2\pi/3 $ radians apart. $\\

$So the other 2 will be $\\

=cos(\pi/3)+isin(\pi/3)\qquad and \qquad cos(-\pi/3)+isin(-\pi/3)\\

=\qquad \frac{1}{2}+\frac{\sqrt3i}{2} \qquad \qquad \qquad and\qquad \qquad \frac{1}{2}-\frac{\sqrt3i}{2}\\

=\qquad \frac{1+\sqrt3i}{2} \qquad \qquad \qquad and\qquad\qquad\frac{1-\sqrt3i}{2}$$

NOW

$$\\\sqrt[3]{-1}=-1,\qquad \frac{1+\sqrt3i}{2}, \qquad\frac{1-\sqrt3i}{2}\\

so\\

\sqrt[3]{14\sqrt[3]{-1}}\\\\

=\sqrt[3]{14(-1)},\qquad \sqrt[3]{14*\frac{1+\sqrt3i}{2}}, \qquad \sqrt[3]{14*\frac{1-\sqrt3i}{2}}\\\\

=\sqrt[3]{-14},\qquad \sqrt[3]{7(1+\sqrt3i)}, \qquad \sqrt[3]{7(1-\sqrt3i)}\\\\

$Now WolframAlpha only gave 3 answers but here I am going to get 3 lots of 3 answers, perhaps each lot is the same as the other 3 lots?$\\\\

\sqrt[3]{-14}\\\\

=\sqrt[3]{14}\sqrt[3]{-1}\\\\

=-\sqrt[3]{14},\qquad \sqrt[3]{14}*\frac{1+\sqrt3i}{2}, \qquad\sqrt[3]{14}*\frac{1-\sqrt3i}{2}\\\\$$

$$\\$This approximates to$\\\\

=-2.410,\qquad 2.410*\frac{1+\sqrt3i}{2}, \qquad2.410*\frac{1-\sqrt3i}{2}\\\\

=-2.410,\qquad 2.410*(0.5+\frac{\sqrt3i}{2}), \qquad2.410*(0.5-\frac{\sqrt3i}{2})\\\\

=-2.410,\qquad 1.205+2.087i, \qquad 1.205-2.087i\\\\$$

Badinage is right. None of these answers agree with Wolfram|alpha

Melody
Apr 21, 2015

#11**+10 **

The best method for dealing with powers and roots of complex numbers, is to switch them to polar form and then to use De Moivre's theorem.

So here,

$$\displaystyle \frac{-7+21\sqrt{3}\imath}{2}=-\frac{7}{2}(1-3\sqrt{3}\imath)=-\frac{7}{2}\sqrt{28}\angle\tan^{-1}(-3\sqrt{3})$$

(ask for details if you need them)

$$\displaystyle = -7\sqrt{7}\angle 280.89339\deg.$$

Raise that to the power one third and you have,

$$(-7\sqrt{7}\angle(-280.89339+360k))^{1/3}$$

$$\displaystyle = -\sqrt{7}\angle(93.63113+120k), \quad k=0,1,2.$$

That gets you

$$\displaystyle k=0:\quad -\sqrt{7}\angle 93.63113 = -\sqrt{7}(\cos93.63113+\imath\sin93.63113)$$

$$\displaystyle =0.16752-2.64044\imath.$$

and the other two values of k,

$$\displaystyle 2.20291+1.46533\imath \;\text{ and } -2.37047+1.17511\imath \quad\text{respectively}.$$

Bertie
Apr 21, 2015

#12**+5 **

**I'd like someone to point out the flaw in fiora's working. I agree there exist more than one answer, but where is the error in fiora's calculation if the result obtained there is not one of the valid answers?**

Badinage
Apr 21, 2015

#13**+10 **

Best Answer

Look at the step where fiora sets -1 + √3i to ^{3}√(-1 + √3i)^{3}

In the next step (-1 + √3i)^{3 }is expanded, correctly, to get 8, but then the wrong cube root of 8 is taken. Fiora says the cube root of 8 is 2. If this were the right thing to do here then it would mean that -1 + √3i equals 2, which would be a little unusual to say the least!

The fact is that, in the domain of complex numbers, there are three cube roots of 8, namely 2, -1 - √3i and -1 + √3i. The only valid root here is the last one. It isn't valid to jump from -1 + √3i to (-1 + √3i)^{3} and then jump back to a different root from the one you came from!

It's like saying -2 squared is 4, the square root of 4 is 2, so we can replace -2 with 2.

.

Alan
Apr 21, 2015