Find all the cube root of −4√2−4√2i
, ,
Let z = −4√2−4√2i
Write given complex number in polar form
|z| = √(−4√2)2+(−4√2)2 = 16
tan−(α) = −4√2−4√2 = 1
α = π4
Since. complex number is in third quadrant
Then, argument θ=−(π−α) = -(π−π4)= −3π4
Polar form of z = 16 [cos(−3π4)+isin(−3π4)]
= 16 (cos3π4−isin3π4)
Cube root of z = 1613[cos(3π4)+isin(−3π4)]13
=1613[cos(2kπ−3π4)3+isin(2kπ−3π4)3]
When k = 0,1,2
When k = 0
1613 [cos(−3π4)3+isin(−3π4)3]
= 1613 [cosπ4−isinπ4]
When k =1
1613 [cos(2π−3π4)3+isin(2π−3π4)3]
1613 [cos5π12+isin5π12]
When k = 2
1613 [cos(4π−3π4)3+isin(4π−3π4)3]
1613 [cos(13π12)+isin13π12]