Torneu a escriure sqrt ((x ^ 2 * 2 ^ x) / ((2x) ^ 3 * 3 ^ (2x) com a * x ^ b * c ^ x per x> 0?

Resposta:

#sqrt ((x ^ 2 * 2 ^ x) / ((2x) ^ 3 * 3 ^ (2x)) = color (vermell) (1 / (2sqrt (2)) * x ^ (color (vermell) (-3/2)) * color (vermell) ((sqrt (2) / 3)) ^ x #

Explicació:

#sqrt ((x ^ 2 * 2 ^ x) / ((2x) ^ 3 * 3 ^ (2x)) #

#color (blanc) ("XXX") = color (blau) (sqrt (x ^ 2) / (sqrt ((2x) ^ 3)) color (verd) (sqrt (2 ^ x) / sqrt (3 ^ (2x))) #

Prenent aquests components un cop (i assumint) #x> 0 #)

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#color (blau) (sqrt (x ^ 2)) = color (marró) x #

#color (blau) (1 / sqrt ((2x) ^ 3) = 1 / ((2x) ^ 3) ^ (1/2) = 1 / ((2x) ^ (3/2)) = 1 / (2 ^ (3/2) * x ^ (3/2)) = 1 / (2sqrt (2) * x ^ (3/2)) = color (marró) (1 / (2sqrt (2)) x ^ (-3/2)) #

#rarr color (blanc) ("XXX") color (blau) (sqrt (x ^ 2) / (sqrt ((2x) ^ 3 * 3 ^ (2x))) = color (marró) x * color (marró)) (1 / (2sqrt (2)) x ^ (- 3/2)) = color (vermell) (1 / (2sqrt (2)) * x ^ (- 1/2) #

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#color (verd) (sqrt (2 ^ x)) = color (marró) (sqrt (2) ^ (x)) #

#color (verd) (1 / sqrt (3 ^ (2x))) = 1 / (3 ^ x) = color (marró) ((1/3) ^ x) #

#rarr color (blanc) ("XXX") color (verd) (sqrt (2 ^ x) / sqrt (3 ^ (2x))) = color (vermell) ((sqrt (2) / 3) ^ x) #