\(12^8\cdot9^{12}=2^{16}\cdot3^8\cdot3^{24}=\left(2\cdot3^2\right)^{16}=18^{16}\)

\(12^8.9^{12}=18^{16}\)

\(\left(3.4\right)^8.3^{24}=\left(2.9\right)^{16}\)

\(3^8.4^8.3^{24}=2^{16}.9^{16}\)

\(3^{32}.2^{16}=2^{16}.3^{32}\)

12⁸.9¹²

=(2².3)⁸.9¹²

=(2²)⁸.3⁸.9¹²

=2¹⁶.(3²)⁴.9¹²

=2¹⁶.9⁴.9¹²

=2¹⁶.9¹⁶

=(2.9)¹⁶

=18¹⁶

12⁸.9¹²=3⁸.4⁸.9¹²=9⁴.4⁸.9¹²=9¹⁶.2¹⁶=18¹⁶

12⁸.9¹²=(2².3)⁸.(3²)¹²=2¹⁶.3⁸.3²⁴=2¹⁶.3³²=2¹⁶.(3²)¹⁶=2¹⁶.9¹⁶=(2.9)¹⁶=18¹⁶

=>12⁸.9¹²=18¹⁶

Lời giải:

$12^8.9^{12}=(2^2.3)^8.(3^2)^{12}=2^{16}.3^8.3^{24}=2^{16}.3^{32}$

$=2^{16}.(3^2)^{16}=2^{16}.9^{16}=(2.9)^{16}=18^{16}$

Ta có \(12^8=\left(12^2\right)^4=144^4\)

         \(9^{12}=\left(9^3\right)^4=729^2\)

=> \(12^8\cdot9^{12}=144^4\cdot729^4=\left(144\cdot729\right)^4=104976^4\)(1)

Ta lại có \(18^{16}=\left(18^4\right)^4=104976^4\)(2)

Từ (1) và (2) => \(12^8\cdot9^{12}=18^{16}\)

12⁸.9¹²

= (3.2.2)⁸.(3²)¹²

= 3⁸.2⁸.2⁸.3²⁴

= 3³².2¹⁶

= (3²)¹⁶.2¹⁶

= 9¹⁶.2¹⁶

= (9.2)¹⁶

= 18¹⁶

=> đpcm.