Kako poenostavite (1 / sqrt (a-1) + sqrt (a + 1)) / (1 / sqrt (a + 1) -1 / sqrt (a-1)) div sqrt (a + 1) / ( (a-1) sqrt (a + 1) - (a + 1) sqrt (a-1)), a> 1?

Kako poenostavite (1 / sqrt (a-1) + sqrt (a + 1)) / (1 / sqrt (a + 1) -1 / sqrt (a-1)) div sqrt (a + 1) / ( (a-1) sqrt (a + 1) - (a + 1) sqrt (a-1)), a> 1?
Anonim

Odgovor:

Ogromno matematično oblikovanje …

Pojasnilo:

#color (modra) (((1 / sqrt (a-1) + sqrt (a + 1)) / (1 / sqrt (a + 1) -1 / sqrt (a-1))) / (sqrt (a +1) / ((a-1) sqrt (a + 1) - (a + 1) sqrt (a-1))) #

# = barva (rdeča) (((1 / sqrt (a-1) + sqrt (a + 1)) / ((sqrt (a-1) -sqrt (a + 1)) / (sqrt (a + 1)) cdot sqrt (a-1)))) / (sqrt (a 1) / (sqrt (a-1) cdot sqrt (a-1) cdot sqrt (a + 1) -sqrt (a + 1) cdot sqrt (a + 1) sqrt (a-1))) #

# = barva (modra) ((((1 / sqrt (a-1) + sqrt (a + 1)) / ((sqrt (a-1) -sqrt (a + 1)) / (sqrt (a + 1)) cdot sqrt (a-1)))) / (sqrt (a 1) / (sqrt (a + 1) cdot sqrt (a-1) (sqrt (a-1) -sqrt (a + 1))) #

# = barva (rdeča) ((1 / sqrt (a-1) + sqrt (a + 1)) / ((sqrt (a-1) -sqrt (a + 1)) / (sqrt (a + 1) cdot sqrt (a-1))) xx (sqrt (a 1) cdot sqrt (a-1) (sqrt (a-1) -sqrt (a + 1))) / sqrt (a + 1) #

# = barva (modra) ((1 / sqrt (a-1) + sqrt (a + 1)) xx ((sqrt (a + 1) cdot sqrt (a-1)) / (sqrt (a-1) - sqrt (a + 1))) xx (prekliči ((sqrt (a + 1))) cdot sqrt (a-1) (sqrt (a-1) -sqrt (a + 1))) / cancelsqrt (a + 1))) #

# = barva (rdeča) (((1 + sqrt (a + 1) cdot sqrt (a-1)) / (sqrt (a-1))) xx ((sqrt (a + 1) cdot sqrt (a-1))) / (sqrt (a-1) -sqrt (a + 1))) xx sqrt (a-1) cdot (sqrt (a-1) -sqrt (a + 1)) #

# = barva (modra) ((((1 + sqrt (a 1) cdot sqrt (a-1)) / prekliči (sqrt (a-1))) xx ((sqrt (+ 1) preklic cdot ((sqrt (a-1)))) / barva (rdeča) (preklic (barva (zelena) ((sqrt (a-1) -sqrt (a + 1))))) xx sqrt (a-1) barva cdot (rdeča)) (prekliči barvo (zelena) ((sqrt (a-1) -sqrt (a + 1))) #

# = color (rdeča) (ul (bar (| color (modra) ((1 + sqrt (a + 1) cdot sqrt (a-1)) cdot (sqrt ((a + 1) (a-1)))) | #

Odgovor:

#sqrt (a ^ 2-1) + a ^ 2-1 #

Pojasnilo:

Za poenostavitev stvari bomo veliko uporabljali # u ^ 2 = a + 1 # in # v ^ 2 = a-1 #, ki nam daje:

# (v ^ -1 + u) / (u ^ -1-v ^ -1) * (uv ^ 2-vu ^ 2) / u = ((v ^ -1 + u) (uv ^ 2-vu ^ 2)) / (u (u ^ -1-v ^ -1)) = (uv-u ^ 2 + (uv) ^ 2-vu ^ 3) / (1-uv ^ -1) = (uv (1) + uv) -u ^ 2 (1 + uv)) / ((vu) / v) = (uv (1 + uv) (vu)) / (vu) = uv (1 + uv) #

#vv (1 + uv) = uv + u ^ 2v ^ 2 = sqrt (a-1) sqrt (a + 1) + (a-1) (a + 1) = sqrt (a ^ 2-1) + a ^ 2-1 #