# Find and graph the cube roots of 216i?

Relevance
• 216i => 216[cos(π/2 + 2nπ) + isin(π/2 + 2nπ)]

(216i)¹​/³ => (216)¹​/³[cos(π/2 + 2nπ) + isin(π/2 + 2nπ)]¹​/³

i.e. 6[cos(π/6 + 2nπ/3) + isin(π/6 + 2nπ/3)]

For n = 0 we have:

6[cos(π/6) + isin(π/6)]

i.e. 6(√3/2 + i/2)

or, 3(√3 + i)

With n = 1 we have:

6[cos(5π/6) + isin(5π/6)]

i.e. 6(-√3/2 + i/2)

or, 3(-√3 + i)

With n = 2 we have:

6[cos(3π/2) + isin(3π/2)]

i.e. 6(0 + -i) => -6i

Hence, 3 cubic roots are -6i, 3(√3 + i) and 3(-√3 + i)

:)>

• Cube root. For this, study 216=3*72= 3*3*24=3*3*3* 8

The cue root of 216 is 6. For i you write i sin pi/2= exp i(2npi +pi/2). The cue root of 216 is 6 exp i(2npi/3+pi/6)

Now write one root, n=0.

r1= root 1= 6 (cos pi/6+i sin pi/6)

r2 (n=1) =6 (exp i( 2pi/3+pi/6)=6 exp i(150). You can expand and get the value

r3 (n=2) 6 exp(i(4pi/3+pi/6))= 6 exp (i270)

These are three roots.

• I'll go first......

1.  convert your  number into trig. form ,   r CiS @     where  r = your number    [  r will always  be > 0,    if  a + bi  given  then  r = sqrt( a^2 + b^2)   ]    and

@ = angle from the + x axis counter clockwise   [ Tan @ = b/a  solve for  @  ]

NOTE:  CiS  does not mean  Crime Investigative Service , like the tv show...it means  Cosine ( )  + i  Sine (  )

216i  =  216( cos 90˚ + i sin 90˚ )    or  216 CiS 90˚    some books call this  z

2.   Use  eqn  for  roots   z ^1/n   =  r^1/n  [  CiS  (   ( @ + 360K ) /  n  )      where  K = 0, 1, 2, ... n - 1      [ in radians  ( @ + 2πK ) / n  ]

so  K = 0, 1, or  2  here   thus  3  roots

216^1/3 = 6

@ + 360K     will =   90 + 360*0 = 90     for K = 0

90 + 360*1 =  450  for  K = 1

and  90 +360*2 = 810    for K = 2

now  divide  by  n,  which is 3  here....  angles  are  90/3 = 30˚   , 450/3 = 150˚ , etc...

so first answer in trig  form is  6 CiS 30˚ ... I'll let you find the others and then convert them into a + bi  form if needed... I hope you know how to do that.