Hvordan forenkler du (1 / sqrt (a-1) + sqrt (a + 1)) / (1 / sqrt (a + 1) -1 / sqrt (a-1)) div sqrt (a-1) sqrt (a + 1) - (a + 1) sqrt (a-1)), a> 1?

Svar:

Kæmpe matematisk formatering …

Forklaring:

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

# = Farve (rød) (((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) sqrt (a-1))) #

# = Farve (blå) (((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)

# = farve (rød) (1 / sqrt (a-1) + sqrt (a + 1)) / ((sqrt (a-1) -sqrt (a + 1)) sqrt (a-1)) xx (sqrt (a + 1) cdot sqrt (a-1) (sqrt (a-1) -sqrt (a + 1))) / sqrt (a + 1)

# = farve (blå) (1 / sqrt (a-1) + sqrt (a + 1)) xx ((sqrt (a + 1) cdot sqrt (a-1)) / sqrt (a + 1))) xx (annuller ((sqrt (a + 1))) cdot sqrt (a-1) (sqrt (a-1) -sqrt (a + 1))) / cancelsqrt)) #

# = farve (rød) ((1 + sqrt (a + 1) cdot sqrt (a-1)) / (sqrt (a-1))) xx ((sqrt (a + 1) cdot sqrt)) / (sqrt (a-1) -sqrt (a + 1)) xx sqrt (a-1) cdot (sqrt (a-1) -sqrt

# = farve (blå) ((1 + sqrt (a + 1) cdot sqrt (a-1)) / annuller (sqrt (a-1))) xx ((sqrt (a + 1) cdot annullere (a-1)))) / farve (rød) (annuller (farve (grøn) ((sqrt (a-1) -sqrt (a + 1)))) xx sqrt (a-1) cdot farve) (annuller farve (grøn) ((sqrt (a-1) -sqrt (a + 1)))) #

# = farve (rød) (ul (bar (| farve (blå) (1 + sqrt (a + 1) cdot sqrt (a-1)) cdot (sqrt ((a + 1) (a-1)))) | #

Svar:

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

Forklaring:

For at forenkle tingene i høj grad vil vi bruge # U ^ 2 = a + 1 # og # V ^ 2 = a-1 #, som giver os:

# (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) #

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