Q13P - Mathematical Methods in Physical Sciences

Question

Follow steps (a), (b), (c) above to find all the values of the indicated roots $\sqrt[\mathbf{4}]{\mathbf{-}\mathbf{4}}$.

Step-by-Step Solution

Answer

The value of $4\sqrt{-4} {\mathrm{is}}^{\pm 1,\pm \mathrm{i}}$.

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

Step 1: Given Information

The given expression is $\sqrt[4]{-4}$. 

Step 2: Definition of Complex Number

Complex numbers consist of real numbers and imaginary numbers; 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 -4+0i .

 

x=-4y, y=0

 

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

 

$$\begin{gathered} r=\sqrt{2} \\ \theta =\pi , or 3\pi ,5\pi ,7\pi 9\pi ,...... \end{gathered}$$

 

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

 

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

When n=4 , the equation becomes the 4th root of the complex number:

 $$\begin{gathered} z^{\frac{1}{4}}=r^{\frac{1}{4}}{e}^{\frac{i\theta }{4}} \\ r=\sqrt{2} \\ \theta =\frac{\pi }{4},\frac{3\pi }{4},\frac{5\pi }{4},\frac{7\pi }{4},\frac{9\pi }{4},....... \end{gathered}$$

Step 4: Plotting the polar coordinate points on the graph

It is clear from the above graph that the points $\left(\sqrt{2},\frac{\pi }{4}\right)$ and the point $\left(\sqrt{2},\frac{9\pi }{4}\right)$are the same.

 

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

$$\begin{gathered} r=\sqrt{2} \\ \theta =\frac{\pi }{4},\frac{3\pi }{4},\frac{5\pi }{4},\frac{7\pi }{4} \end{gathered}$$

 

Hence, the value of $\sqrt[4]{-4}=\pm 1,\pm \mathrm{i}$  .