Question

If ${\text{z = }}\dfrac{{{\text{1 + i}}}}{{\sqrt {\text{2}} }}$ ,then the value of ${{\text{z}}^{1929}}$ is
A. $1 + i$
B. -1
C. $\dfrac{{{\text{1 + i}}}}{{\text{2}}}$
D. $\dfrac{{{\text{1 + i}}}}{{\sqrt {\text{2}} }}$

Answer 1

Here we have to use the idea of $|z|$ and argument of z . Than use the concept of writing of $z = x + iy$ as $|z|{e^{i\theta }},$ where $\theta = {\tan ^{ - 1}}\dfrac{y}{x}$ . And then proceed with dividing the angle and apply general trigonometry.

Complete step by step answer:

As per the given equation, $z = \dfrac{{1 + i}}{{\sqrt 2 }}$
We have $x = \dfrac{1}{{\sqrt 2 }}$ and $y = \dfrac{1}{{\sqrt 2 }}$ ,
As, $|z| = \sqrt {{x^2} + {y^2}}$
On substituting values of x and y we get,
$\Rightarrow$ ${\text{|z}}| = \sqrt {{{(\dfrac{1}{{\sqrt 2 }})}^2} + {{(\dfrac{1}{{\sqrt 2 }})}^2}}$
On simplification we get,
$\Rightarrow$ ${\text{|z}}| = \sqrt {\dfrac{1}{2} + \dfrac{1}{2}}$
$\Rightarrow$ ${\text{|z}}| = \sqrt 1$
On taking positive square root we get,
$\Rightarrow$ $|z| = 1$
Now proceeding with the calculation of the argument of z.
${\text{$\theta$ = ta}}{{\text{n}}^{{\text{ - 1}}}}\dfrac{{\text{y}}}{{\text{x}}}$
On substituting the value of x and y we get,
$\Rightarrow \theta = {\text{ta}}{{\text{n}}^{{\text{ - 1}}}}\dfrac{{\dfrac{1}{{\sqrt 2 }}}}{{\dfrac{1}{{\sqrt 2 }}}}$
On simplification we get,

$$\Rightarrow \theta = {\tan ^{ - 1}}(1) \\\ \Rightarrow \theta = \dfrac{\pi }{4} \\\$$

Hence, the given equation can also be converted into the form of ${\text{|z|}}{{\text{e}}^{{\text{${i\theta}$ }}}}$,
$\Rightarrow z$ = ${\text{|z|}}{{\text{e}}^{{\text{${i\theta}$ }}}}$

$$\Rightarrow {\text{z = }}{{\text{e}}^{{\text{i}}\dfrac{{\text{$\pi$ }}}{{\text{4}}}}} \\\$$

So,
$\Rightarrow {{\text{z}}^{{\text{1929}}}}{\text{ = }}{{\text{e}}^{{\text{(1929)i}}\dfrac{{\text{$\pi$ }}}{{\text{4}}}}}$
Now as $1929(\dfrac{\pi }{4}) = 482\pi + \dfrac{\pi }{4}$, so we get,
$\Rightarrow {{\text{z}}^{{\text{1929}}}}{\text{ = }}{{\text{e}}^{{\text{i(482$\pi$ + }}\dfrac{{\text{$\pi$ }}}{{\text{4}}}{\text{)}}}}$
Now as $482\pi + \dfrac{\pi }{4} = \dfrac{\pi }{4}$, so we get,
$\Rightarrow {{\text{z}}^{{\text{1929}}}}{\text{ = }}{{\text{e}}^{{\text{i}}\dfrac{{\text{$\pi$ }}}{{\text{4}}}}}$
As we have ${\text{z = }}{{\text{e}}^{{\text{i}}\dfrac{{\text{$\pi$ }}}{{\text{4}}}}}$
$\Rightarrow {{\text{z}}^{{\text{1929}}}}{\text{ = z = }}\dfrac{{{\text{1 + i}}}}{{\sqrt {\text{2}} }}$
Hence, option (d) is our correct answer.

Note: A complex number is a number that can be expressed in the form $a{\text{ }} + {\text{ }}bi$, where a and b are real numbers, and i represents the imaginary unit. Because no real number satisfies this equation, i is called an imaginary number. Where $\theta = \arg z$ and so we can state that, much like the polar form, there are an infinite number of possible exponential forms for a given complex number. Also, because any two arguments for a give complex number differ by an integer multiple of 2$\pi$ . we will sometimes write the exponential form as, z = r{e^{i(\theta + 2$\pi$ n)}},n = \pm 1, \pm 2...