Question

How do you find the fifth roots of$128( - 1 + i)$?

Answer 1

Hint : Convert the given complex number into its polar form for easy calculation and use de Moivre’s theorem.
The solving of the complex number and finding its roots takes place in the polar form majorly. Because in this form we can easily convert the values to trigonometric values at our convenience.
The polar form of a complex number $z = a + ib$is $\theta = r(\cos \theta + i\sin \theta )$ , where $r = |z| = \sqrt {{a^2} + {b^2}}$, $a = rcos\theta$ and $b = rsin\theta$ , and $\theta = ta{n^{ - 1}}\left( {ba} \right)$ for a>0 and $\theta = ta{n^{ - 1}}\left( {ba} \right) + \pi$ or $\theta = ta{n^{ - 1}}\left( {ba} \right) + 180^\circ$ for a<0 .

Complete step-by-step answer :
Multiply and divide by $\sqrt 2$for conversion into trigonometric form.
And upon conversion calculate further.
$128( - 1 + i) = 128\sqrt 2 ( - \dfrac{1}{{\sqrt 2 }} + i(\dfrac{1}{{\sqrt 2 }})) \\\ = 128\sqrt 2 ( - \cos (\dfrac{\pi }{4}) + i\sin (\dfrac{\pi }{4})) \;$
We know that the cos function is negative and sin function is positive in the second quadrant which means that we can also write the above as,
$= 128\sqrt 2 (\cos (135) + i\sin (135))$
The value of $\cos (\dfrac{\pi }{4}),\sin (\dfrac{\pi }{4})$ are $\dfrac{1}{{\sqrt 2 }}$.
For finding the fifth root we have to do the following process:
$\sqrt[5] {{128\sqrt 2 (\cos 135 + i\sin 135)}}$
Now, we are going to apply de Moivre’s theorem, which is given below
${\left[ {r(\cos \theta + i\sin \theta } \right] ^n} = {r^n}\left[ {\cos n\theta + i\sin n\theta } \right]$
We, will get
$= {(128\sqrt 2 )^{\dfrac{1}{5}}}(\cos (\dfrac{{135}}{5}) + i\sin (\dfrac{{135}}{5})) \\\ = (\sqrt 8 (\cos 27 + i\sin 27) \;$
In the above equation, it was simplified and written by,
$= {(128\sqrt 2 )^{\dfrac{1}{5}}} \\\ = {({2^7} \times \sqrt 2 )^{\dfrac{1}{5}}} \\\ = {(\sqrt 2 )^{\dfrac{{15}}{5}}} \\\ = \sqrt 8 \;$
In the above process the multiplication of exponential powers is done for getting the desired value or simplified value easily.
Without the exponential multiplication or exponential operations solving of the problems would be tuff hence, we use exponential operations.
Hence, the value of the fifth root of the complex number is $(\sqrt 8 *(\cos 27 + i\sin 27))$.
So, the correct answer is “ $(\sqrt 8 *(\cos 27 + i\sin 27))$ ”.

Note : While solving the above problem many students face the issue in conversion of complex form into polar form. For that one has to have a clear idea about the trigonometric values and their operations. The trigonometric values also should be equal for both sin and cosine so that the conversion would be easier for calculation. Hence, the students should have a clear idea about trigonometry, exponential operations and also the complex numbers for easy solving of the problem.