Root sum

What is the sum of the fifth root of 243?

Final Answer:

s = 0

a = 5243 = 3 α = \frac{2 π}{5} = \frac{2 \cdot 3 . 1 4 1 6}{5} ≐ 1.2566 rad x_{1} = a \cdot cos (0 \cdot α) = 3 \cdot cos (0 \cdot 1.2566) = 3 x_{2} = a \cdot cos (1 \cdot α) = 3 \cdot cos (1 \cdot 1.2566) ≐ 0.9271 x_{3} = a \cdot cos (2 \cdot α) = 3 \cdot cos (2 \cdot 1.2566) ≐ - 2.4271 x_{4} = a \cdot cos (3 \cdot α) = 3 \cdot cos (3 \cdot 1.2566) ≐ - 2.4271 x_{5} = a \cdot cos (4 \cdot α) = 3 \cdot cos (4 \cdot 1.2566) ≐ 0.9271 x = x_{1} + x_{2} + x_{3} + x_{4} + x_{5} = 3 + 0.9271 + (- 2.4271) + (- 2.4271) + 0.9271 = 0 y_{1} = a \cdot sin (0 \cdot α) = 3 \cdot sin (0 \cdot 1.2566) = 0 y_{2} = a \cdot sin (1 \cdot α) = 3 \cdot sin (1 \cdot 1.2566) ≐ 2.8532 y_{3} = a \cdot sin (2 \cdot α) = 3 \cdot sin (2 \cdot 1.2566) ≐ 1.7634 y_{4} = a \cdot sin (3 \cdot α) = 3 \cdot sin (3 \cdot 1.2566) ≐ - 1.7634 y_{5} = a \cdot sin (4 \cdot α) = 3 \cdot sin (4 \cdot 1.2566) ≐ - 2.8532 y = y_{1} + y_{2} + y_{3} + y_{4} + y_{5} = 0 + 2.8532 + 1.7634 + (- 1.7634) + (- 2.8532) = 1.0036 \cdot 1 0^{- 13} s = x + i \cdot y s = 0

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