Question

The complex numbers u,v and w are related by 1/u=1/v+2/w.If v=3-4i and w=4+3i find u in the rectangular form​

Answer 1

Step-by-step explanation:

Consider the complex numbers u = 2 + 3i and v = 3 + 2i. a) Given that \frac{1}{u} + \frac{1}{v} =...

Question:

Consider the complex numbers {eq}\displaystyle u = 2 + 3i {/eq} and {eq}\displaystyle v = 3 + 2i {/eq}.

a) Given that {eq}\displaystyle \frac{1}{u} + \frac{1}{v} = \frac{10}{w}, {/eq} express w in the form {eq}\displaystyle a + bi, a, b \in R. {/eq}

b) Find {eq}\displaystyle w^8 {/eq} and express it in the form {eq}\displaystyle re^{i\theta}. {/eq}

De Moivre's Theorem of Complex Numbers:

Suppose we have a complex number in the polar form such as {eq}z=r(\cos \theta +i\sin \theta ) {/eq}, then we can find the value of the complex number with exponent {eq}n {/eq} by De Moivre's theorem as:

{eq}z^n=r^n(\cos \theta +i\sin \theta )^n\\ =r^n(\cos n \theta +i\sin n\theta )\\ {/eq}

The conversation of polar coordinates from the cartesian coordinates and the exponential form from polar expression is obtained as:

{eq}\displaystyle r=\sqrt{x^2+y^2}\\ \theta =\tan^{-1}\frac{y}{x}\\ r(\cos \theta +i\sin \theta)=re^{i\theta} {/eq}

Answer and Explanation: