Powers of complex numbers calculator

SolutionHelp

(5+6i),(-2+3i),(1-3i) $(5+6i), (-2+3i),(1-3i)$

(3-2i),(4+3i),(2-5i) $(3-2i), (4+3i),(2-5i)$

Example

1. A=5+6i,B=-2+3i,C=1-3i $A=5+6i,B=-2+3i,C=1-3i$
Find pow(A,6)

Solution:
Here

A=5+6i,B=-2+3i,C=1-3i $A=5+6i,B=-2+3i,C=1-3i$

For a complex number

z=a+bi $z=a+bi$ , the polar form is

z=r⋅(cos(θ)+i⋅sin(θ)) $z=r*(cos(theta)+i*sin(theta))$

then power of n of given complex number can be obtained by

zn=[r⋅(cos(θ)+i⋅sin(θ))]n=rn⋅[cos(n⋅θ)+i⋅sin(n⋅θ)] $z^n=[r*(cos(theta)+i*sin(theta))]^n=r^n*[cos(n*theta)+i*sin(n*theta)]$

Step-1: Convert to exponential form: z=reiθ $z = re^(i theta)$

Here,

a=5 $a=5$ and

b=6 $b=6$

$:. r=sqrt(5^2+6^2)=sqrt(25+36)=sqrt(61)=7.8102$

$theta=atan(b/a)$ (Since

$a>0$ )

$:. theta=atan((6)/(5))$

$:. theta=atan(1.2)$

$:. theta=50.1944 ^circ$ or

$theta=0.8761$ rad

$:. theta=0.8761$

Exponential form:

$5+6i=r*e^(i*theta)$

$5+6i=7.8102*e^(i(0.8761))$

Step-2: Apply the power formula
Now

$(5+6i)^(6)=(7.8102)^(6)*e^(i(6*0.8761))$

$=226981*e^(i(5.2563))$

Step-3: Convert back to rectangular form

$=226981*(cos(5.2563)+isin(5.2563))$

$=226981*(0.5175-0.8557i)$

$=117469-194220i$