Final answer:
To find the values of (10i/(1+i))1/6, we simplify the fraction, convert it to polar form, and then use De Moivre's theorem to obtain six complex roots in polar form, which can be converted to rectangular form.
Step-by-step explanation:
To find all the values of (10i/(1+i))1/6, we first need to simplify the fraction inside the parenthesis before applying De Moivre's theorem for finding the roots of complex numbers.
- Rewrite and simplify the fraction:
10i/(1+i) = 10i/(1+i) × (1-i)/(1-i) = (10i - 10)/(1 - i²) = (10i - 10)/2 = 5i - 5 - Convert to polar form (use r for modulus and θ for argument):
r = √((5)² + (-5)²) = √(50), θ = arctan(-5/5) = arctan(-1) = -π/4 (or 7π/4 in the range [0, 2π]) - Apply De Moivre's theorem to find the 6th roots:
The 6th roots will have a modulus of √(50)1/6 and arguments of (-π/4 + 2kπ)/6 for k=0,1,2,3,4,5. - List all roots using the polar to rectangular conversion:
The final answers would be a list of six complex numbers derived from the polar forms calculated above.