Question

One value of $(1+i{)}^{1/2}$ is $2^{1/4}{e}^{i\pi /8}$. The other value is

• A. $2^{1/4}{e}^{-i\left(\pi /8\right)}$
• B. $2^{1/4}{e}^{i\left(5\pi /8\right)}$
• C. $2^{1/4}{e}^{-i\left(5\pi /8\right)}$
• D. $2^{1/4}.e^{i\left(9\pi /8\right)}$

Answer 1

The correct option is D $2^{1/4}.e^{i\left(9\pi /8\right)}$

$(1+i)=\sqrt{2}{e}^{\frac{\pi }{4}}$

$\theta =\frac{\pi }{4}$

$\text{We rotate}(1+i)\text{with}2\pi \text{. so that}$

$\text{we can reach at same coordinate}$

$\theta =\frac{\pi }{4}+2\pi$

$\theta =\frac{9\pi }{4}$

We can also write $(1+i)$ as $\sqrt{2}{e}^{\frac{9\pi }{4}}$

$(1+i{)}^{\frac{1}{2}}={(2^{\frac{1}{2}}\cdot e^{\frac{9\pi }{4}})}^{\frac{1}{2}}$

$={\left(2^{\frac{1}{2}}\right)}^{\frac{1}{2}}\cdot {\left(e^{\frac{9\pi }{4}}\right)}^{\frac{1}{2}}$

$=2^{\frac{1}{4}}\cdot e^{\frac{9\pi }{8}}$

option $D$ is correct.