Q17P - Mathematical Methods in Physical Sciences

Question

Follow steps (a), (b), (c) above to find all the values of the indicate droots $\sqrt[\mathbf{5}]{\mathbf{-}\mathbf{1}}$

Step-by-Step Solution

Answer

The value of $\sqrt[5]{-1} is -1,0.809\pm 0.588i,-0.309\pm 0.951i$.

The graph used in this question to find the answer is shown below.

Step 1: Given Information

The given expression is $\sqrt[5]{-1}$.

Step 2: Definition of Complex Number

Complex numbers have both real numbers and imaginary numbers in them; a complex  can be written in the form of: 

 

z=a+ib

 

Here a and b are real numbers, and i is the imaginary number which is known as iota, whose value is $\sqrt{-1}$.

Step 3: Find the value of r and θ

The Complex number is in the form -1+0i.

 

x=-1,y=0

 

The polar coordinates of the point are in the form of $z=r{e}^{i\theta }$

r=1

$\theta =\pi ,or 3\pi ,5\pi ,7\pi ,9\pi ,11\pi ,....$ 

The equation $z=r{e}^{i\theta }$can also be written in another form.

 

$$\begin{gathered} z^{\frac{1}{n}}={\left(r{e}^{i\theta }\right)}^{\frac{1}{n}} \\ =r{e}^{\frac{1}{n}}{e}^{\frac{i\theta }{n}} \\ {z}^{\frac{1}{n}}=\sqrt[n]{r}\left(\cos \frac{\theta }{n}+i \sin \frac{\theta }{n}\right) \end{gathered}$$

 

When n=5 , the equation becomes the 5th root of the complex number:

 $$\begin{gathered} z^{\frac{1}{5}}=r^{\frac{1}{5}}{e}^{\frac{i\theta }{5}} \\ r=1 \\ \theta =\frac{\pi }{5},\frac{3\pi }{5},\frac{5\pi }{5},\frac{7\pi }{5},\frac{9\pi }{5},\frac{11\pi }{5},..... \\ =\frac{\pi }{5},\frac{3\pi }{5},\pi ,\frac{7\pi }{5},\frac{9\pi }{5},\frac{11\pi }{5},..... \end{gathered}$$

Step 4: Plotting the polar coordinate points on the graph

It is clear from the above graph that the points $\left(1,\frac{\pi }{5}\right)$ and the point $\left(1,\frac{11\pi }{5}\right)$are the same.

The radius of the circle 1 is and equally spaced$\frac{2\pi }{5}$ apart.

$$\begin{gathered} r=1 \\ \theta =\frac{\pi }{5},\frac{3\pi }{5},\pi ,\frac{7\pi }{5},\frac{9\pi }{5} \end{gathered}$$

 

Hence, the value of  $\sqrt[5]{-1}=-1,0.809\pm 0.588i,-0.309\pm 0.951i$