Wenn z = -1 - i ist, finden Sie z10 in polarer Form?

Antworten:

# (- 1 -i) ^ {10} = 32 (cos (pi / 2) + i sin (pi / 2)) = 32 i #

Erläuterung:

#z = -1 -i = sqrt {2} (- 1 / sqrt {2} -i 1 / sqrt {2}) = sqrt {2} (cos ({5pi} / 4) + i sin ({5 pi} / 4)) #

# z ^ {10} = (sqrt {2} (cos ({5pi} / 4) + i sin ({5 pi} / 4))) {10} #

# = (sqrt {2}) ^ {10} (cos ({50 pi} / 4) + i sin ({50 pi} / 4)) #

# = 2 ^ 5 (cos ({25 pi} / 2 - 12 pi) + i sin ({25 pi} / 2 - 12 pi)) #

# = 32 (cos (pi / 2) + i sin (pi / 2)) #

Das ist die Antwort in polarer Form, aber wir machen den nächsten Schritt.

# z ^ {10} = 32 i #