Question

How do you find the 5th root of $1-i$?

Answer 1

To find the nth root of a complex number $Z=a+ib$. We need to follow the steps in the given order. The first step is to convert the complex number to its trigonometric form. Find the appropriate angle. Assume a constant number $\alpha$ such that ${{\alpha }^{n}}=Z$. Take nth root of both sides, and make required changes to the right-hand side.

Complete step-by-step answer:
We are given the complex number $1-i$, we name this number as Z. To solve this problem, we first need to express Z in trigonometric form. To do this, we calculate modulus of Z as, $\left| Z \right|=\sqrt{1+1}=\sqrt{2}$. Using this we can write the trigonometric form of Z as $Z=\sqrt{2}\left( \dfrac{1}{\sqrt{2}}-\dfrac{1}{\sqrt{2}}i \right)$.
We need to find an angel such that $\cos \theta =\dfrac{1}{\sqrt{2}}\And \sin \theta =-\dfrac{1}{\sqrt{2}}$, in the range of $\left[ -\pi ,\pi \right]$. We know that there is only one such angle in the range that satisfies the above condition, $\theta =-\dfrac{\pi }{4}$.
So, using this angle we can simplify the trigonometric form as $Z=\sqrt{2}\left( \cos \left( -\dfrac{\pi }{4} \right)+i\sin \left( -\dfrac{\pi }{4} \right) \right)$.
We can express this complex number in the exponential form as ${{e}^{-\dfrac{\pi }{4}i}}$.
We are asked to find 5th root of Z. Say there exist a complex number $\alpha$that satisfies the condition ${{\alpha }^{5}}=Z$. So, we need to find this complex number $\alpha$.
${{\alpha }^{5}}=Z$
${{\alpha }^{5}}=\sqrt{2}{{e}^{-\dfrac{\pi }{4}i}}$
Taking 5th root of both sides of above equation, we get
${{\left( {{\alpha }^{5}} \right)}^{\dfrac{1}{5}}}={{\left( \sqrt{2}{{e}^{-\dfrac{\pi }{4}i}} \right)}^{\dfrac{1}{5}}}$
Simplifying the above equation, we get $\alpha ={{\left( \sqrt{2} \right)}^{\dfrac{1}{5}}}{{e}^{-\dfrac{\pi }{20}i}}$. We can express it into trigonometric form as $Z={{\left( \sqrt{2} \right)}^{\dfrac{1}{5}}}\left( \cos \left( -\dfrac{\pi }{20} \right)+i\sin \left( -\dfrac{\pi }{20} \right) \right)$.

Note: To solve these types of questions, we just have to follow the above steps. One should know the value of trigonometric ratios of different angles for these types of questions. Calculation mistakes should be avoided.