Complex Numbers

To find one of the values of (−64i)5∕6, let's proceed with the computation:
Let Z=(−64i)5∕6.
We can express Z as:
Z=((−i)5)1∕6×(64)5∕6
Calculate (64)5∕6 :
645∕6=32
Handle the imaginary unit:
Since (−i)5=−i, we represent this in polar coordinates:
−i=cos(‌

3π

2

)+isin‌(‌

3π

2

)
Find (−i)1∕6 :
Using De Moivre's theorem, (−i)1∕6 can be expressed as:
‌(−i)1∕6=[cos(‌

3π

2

)+isin‌(‌

3π

2

)]1∕6
‌=cos(‌

3π

12

)+isin‌(‌

3π

12

)
‌=‌

1

√2

(1+i)
Combine the results:
Therefore, Z becomes:
Z=32×‌

1

√2

(1+i)=16√2(1+i)
The calculation yields one of the values of (−64i)5∕6 as 16√2(1+i).