Ano ang ibig sabihin ng (1-3i) / sqrt (1 + 3i)?

Ano ang ibig sabihin ng (1-3i) / sqrt (1 + 3i)?
Anonim

Sagot:

# (1-3i) / sqrt (1 + 3i) #

# = (- 2sqrt ((sqrt (10) +1) / 2) + 3 / 2sqrt ((sqrt (10) -1) / 2)) - (2sqrt ((sqrt (10) 3 / 2sqrt ((sqrt (10) +1) / 2)) i #

Paliwanag:

Sa pangkalahatan ang square roots ng # a + bi # ay:

# + - ((sqrt ((sqrt (a ^ 2 + b ^ 2) + a) / 2)) + (b / abs (b) sqrt ((sqrt (a ^ 2 + b ^ 2) 2)) i) #

Tingnan ang:

Sa kaso ng # 1 + 3i #, pareho ang Real at haka-haka na mga bahagi ay positibo, kaya nasa Q1 at may isang mahusay na tinukoy na punong square root:

#sqrt (1 + 3i) #

# = sqrt ((sqrt (1 ^ 2 + 3 ^ 2) +1) / 2) + sqrt ((sqrt (1 ^ 2 + 3 ^ 2) -1)

# = sqrt ((sqrt (10) +1) / 2) + sqrt ((sqrt (10) -1) / 2) i #

Kaya:

# (1-3i) / sqrt (1 + 3i) #

# = ((1-3i) sqrt (1 + 3i)) / (1 + 3i) #

# = ((1-3i) ^ 2 sqrt (1 + 3i)) / ((1 + 3i) (1-3i)) #

# = ((1-3i) ^ 2 sqrt (1 + 3i)) / 4 #

# = 1/4 (1-3i) ^ 2 (sqrt ((sqrt (10) +1) / 2) + sqrt ((sqrt (10) -1) / 2) i)

# = 1/4 (-8-6i) (sqrt ((sqrt (10) +1) / 2) + sqrt ((sqrt (10) -1) / 2) i)

# = - 1/2 (4 + 3i) (sqrt ((sqrt (10) +1) / 2) + sqrt ((sqrt (10) -1) / 2) i)

# = - 1/2 ((4sqrt ((sqrt (10) +1) / 2) -3sqrt ((sqrt (10) -1) / 2)) + (4sqrt ((sqrt (10) -1) / 2) + 3sqrt ((sqrt (10) +1) / 2)) i) #

# = (- 2sqrt ((sqrt (10) +1) / 2) + 3 / 2sqrt ((sqrt (10) -1) / 2)) - (2sqrt ((sqrt (10) 3 / 2sqrt ((sqrt (10) +1) / 2)) i #