Q15P - Mathematical Methods in Physical Sciences

Question

Follow steps (a), (b), and (c) above to find all the values of the indicate droots $\sqrt[\mathbf{6}]{\mathbf{-}\mathbf{64}}$.

Step-by-Step Solution

Answer

The value of $\sqrt[6]{-64} is \pm 2i,\pm \sqrt{3}\pm i$.

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

Step 1: Given Information

 The given expression is $\sqrt[6]{-64}$.

Step 2: Definition of Complex Number

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

 

$\mathrm{z}=\mathrm{a}+\mathrm{ib}$

 

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 -64+0i .

 

x=-64, y=0

 

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

 

$$\begin{gathered} r=64 \\ \theta =\pi , or 3\pi ,7\pi ,9\pi ,11\pi ,13\pi ,.... \end{gathered}$$

 

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

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

 

When n=6, the equation becomes the 6th root of the complex number.

$$\begin{gathered} z^{\frac{1}{6}}=r^{\frac{1}{6}}{e}^{\frac{i\theta }{6}} \\ r=2 \\ \theta =\frac{\pi }{6},\frac{3\pi }{6},\frac{5\pi }{6},\frac{7\pi }{6},\frac{9\pi }{6},\frac{11\pi }{6},\frac{13\pi }{6},...... \\ =\frac{\pi }{6},\frac{\pi }{2},\frac{5\pi }{6},\frac{7\pi }{6},\frac{3\pi }{6},\frac{11\pi }{6},\frac{13\pi }{6},...... \end{gathered}$$ 

Step 4: Plotting the polar coordinate points on the graph.

It is clear from the above graph that the points $\left(2,\frac{\pi }{6}\right)$and the point $\left(2,\frac{13\pi }{6}\right)$ are the same.

The radius of the circle is and equally spaced $\frac{\pi }{3}$apart.

r=2

$\theta =\frac{\pi }{6},\frac{\pi }{2},\frac{5\pi }{6},\frac{7\pi }{6},\frac{3\pi }{6},\frac{11\pi }{6}$