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

Hint: 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, \[z = r{e^{i(\theta + 2$\pi$ n)}},n = \pm 1, \pm 2...\]