Question

Write ${\left( {{i^{25}}} \right)^3}$ in polar form.

Answer 1

Hint: We can simplify the given complex number using the powers of i. We can expand the powers of I and simplify it using the relation ${i^{4n + r}} = {i^r}$ . After simplification, we can find its modulus and its argument $\theta $ . Then we can express the complex number in the form $z = r\left( {\cos \theta + i\sin \theta } \right)$

Complete step-by-step answer:
Let $z = {\left( {{i^{25}}} \right)^3}$
We know that, ${\left( {{a^b}} \right)^c} = {a^{b \times c}}$
  $ \Rightarrow z = {i^{25 \times 3}}$
On simplification we get,
 $ \Rightarrow z = {i^{75}}$
Now we can write 75 as the product of 4 and remainder.
 $75 = 4 \times 18 + 3$
Using this we get,
 $ \Rightarrow z = {i^{4 \times 18 + 3}}$
As, \[{a^{b + c}} = {a^b}{a^c}\] and ${\left( {{a^b}} \right)^c} = {a^{b \times c}}$ , using this we get,
 $ \Rightarrow z = {\left( {{i^4}} \right)^{18}} \times {i^3}$
We know that ${i^4} = 1$
 $ \Rightarrow z = {1^{18}} \times {i^3}$
Expanding the power of i, we get,
 $ \Rightarrow z = {i^{2 + 1}}$
Using \[{a^{b + c}} = {a^b}{a^c}\] , we get,
 $ \Rightarrow z = {i^2} \times i$
We know that ${i^2} = - 1$
 $ \Rightarrow z = - i$
Now we have the complex number in the simplest form.
Now we can find its modulus. We know that modulus of a complex number $z = x + iy$ is given by,
 $r = \sqrt {{x^2} + {y^2}} $
On substituting the values, we get,
 $ \Rightarrow r = \sqrt {{0^2} + {{\left( { - 1} \right)}^2}} $
On simplification we get,
 $ \Rightarrow r = 1$
We know that argument of a complex number is given by,
 $\theta = {\tan ^{ - 1}}\left( {\dfrac{y}{x}} \right)$
On substituting the value, we get,
 $ \Rightarrow \theta = {\tan ^{ - 1}}\left( {\dfrac{{ - 1}}{0}} \right)$
We know that $\tan \dfrac{{ - \pi }}{2} = \dfrac{{ - 1}}{0}$
 $ \Rightarrow \theta = \dfrac{{ - \pi }}{2}$
Now we have the argument and modulus. We know that the complex number in polar form is given by, $z = r\left( {\cos \theta + i\sin \theta } \right)$
On substituting the values, we get,
 $z = 1\left( {\cos \dfrac{{ - \pi }}{2} + i\sin \dfrac{{ - \pi }}{2}} \right)$
Therefore, the required polar form is $z = 1\left( {\cos \dfrac{{ - \pi }}{2} + i\sin \dfrac{{ - \pi }}{2}} \right)$.

Note: We know that complex numbers can be plotted in a plane. The complex number $z = x + iy$ is represented in its polar form as $z = r\left( {\cos \theta + i\sin \theta } \right)$ where r is the modulus and $\theta $ is the argument of the complex number.
Modulus of a complex number is the distance from the origin to the complex number in a plane. It is given by the equation $r = \sqrt {{x^2} + {y^2}} $ . Argument of the complex number is the angle that the modulus of the complex number makes with the positive x axis. The angle measured in counter-clockwise direction is positive and angle measured in clockwise direction is negative.