378 x 24352 = 378 x ( 35 )52 = 378 x 3260 = 3338
9101 x 2778 = ( 32 )101 x ( 33 )78 = 3202 x 3234 = 3436
Vì 338 < 436
=> 3338 < 3436
=> 378 x 24352 < 9101 x 2778
Ta có:
\(3^{78}\times243^{52}=3^{78}\times\left(3^5\right)^{52}=3^{78}\times3^{260}=3^{338}\)
\(9^{101}\times27^{78}=\left(3^2\right)^{101}\times\left(3^3\right)^{78}=3^{202}\times3^{234}=3^{436}\)
Vì \(3^{338}< 3^{436}\)
\(\Rightarrow3^{338}< 3^{436}\)
\(\Rightarrow3^{78}\times243^{52}< 9^{101}\times27^{78}\left(đpcm\right)\)