Apakah punca kuasa dua kuasa 27 hingga kuasa 3?

Jawapan:

#sqrt (27) ^ 3 = sqrt (27 ^ 3) = sqrt (3 ^ 9) = 3 ^ (9/2) = 3 ^ 4 3 ^ (1/2) = 81sqrt (3)

Penjelasan:

Gunakan identiti berikut (#a, b, c> = 0 #):

#sqrt (a) = a ^ (1/2) #

# (a ^ b) ^ c = a ^ (bc) #

# a ^ (b + c) = a ^ b a ^ c #

Oleh kerana persoalannya sedikit samar, saya mula-mula menunjukkan bahawa kedua-dua makna yang mungkin bersesuaian:

#sqrt (27) ^ 3 = sqrt (27) sqrt (27) sqrt (27) = sqrt (27 * 27 * 27) = sqrt (27 ^ 3)

Sekarang #27 = 3^3#, jadi

#sqrt (27 ^ 3) = sqrt ((3 ^ 3) ^ 3) = sqrt (3 ^ (3 * 3)) = sqrt (3 ^ 9) #

Kemudian:

= 3 ^ (9 * 1/2) = 3 ^ (9/2) = 3 ^ (4 + 1/2) = 3 ^ 4 3 ^ (1/2) = 81sqrt (3) #

Jadi: #sqrt (27) ^ 3 = sqrt (27 ^ 3) = 81sqrt (3) #