Resposta:
El valor és #-2#
Explicació:
Deixar # x = (2 + sqrt5) ^ (1/3) + (2-sqrt5) ^ (1/3) # llavors
# x ^ 3 = {(2 + sqrt5) ^ (1/3) + (2-sqrt5) ^ (1/3)} ^ 3 #
Recordatori:
# (a + b) ^ 3 = a ^ 3 + b ^ 3 + 3ab (a + b), a ^ 2-b ^ 2 = (a + b) (a-b) #
i ho deixem # a = (2 + sqrt5), b = (2-sqrt5):. ab = 4-5 = -1 #
#:. x ^ 3 = (2 + sqrt5) ^ (3 * 1/3) + (2-sqrt5) ^ (3 * 1/3) +3 (2 + sqrt5) (2-sqrt5) (2 + sqrt5 + 2-sqrt5) #
o bé # x ^ 3 = 2 + cancel·lar (sqrt5) + 2-cancel (sqrt5) +3 (4-5) (2 + cancel (sqrt5) + 2-cancel (sqrt5)) #
o bé # x ^ 3 = 4 + 3 (-1) (4) o x ^ 3 = 4 -12 o x ^ 3 = -8 # o bé
#x = (-8) ^ (1/3) = -2. El valor és #-2# Ans
