Jawapan:
Nilai itu ialah #-2#
Penjelasan:
Biarkan # x = (2 + sqrt5) ^ (1/3) + (2-sqrt5) ^ (1/3) # kemudian
# x ^ 3 = {(2 + sqrt5) ^ (1/3) + (2-sqrt5) ^ (1/3)} ^ 3 #
Peringatan:
# (a + b) ^ 3 = a ^ 3 + b ^ 3 + 3ab (a + b), a ^ 2-b ^ 2 = (a + b) (a-b)
dan biarkan # a = (2 + sqrt5), b = (2-sqrt5):. ab = 4-5 = -1 #
(3 + 1/3) + (2-sqrt5) ^ (3 * 1/3) +3 (2 + sqrt5) (2-sqrt5) (2 + sqrt5 + 2-sqrt5) #
atau # x ^ 3 = 2 + cancel (sqrt5) + 2-cancel (sqrt5) +3 (4-5) (2 + cancel (sqrt5) + 2-cancel (sqrt5)
atau # x ^ 3 = 4 + 3 (-1) (4) atau x ^ 3 = 4 -12 atau x ^ 3 = -8 # atau
#x = (-8) ^ (1/3) = -2 #. Nilai itu ialah #-2# Ans
