a) 27* : 3* = 9

(27 : 3)* = 9

9* = 9

* = 1

Vậy * = 1

b) 81 : (-3)* = -243

(-3)⁴ : (-3)* = (-3)⁵

(-3)* = (-3)⁴ : (-3)⁵

(-3)* = (-3)^-1

* = -1

Vậy * = -1

c) \(\dfrac{1}{2}.\) 2* + 4.2* = 9.2⁵

2*(\(\dfrac{1}{2}\) + 4) = 288

2* . \(\dfrac{9}{2}\) = 288

2* = 288 : \(\dfrac{9}{2}\)

2* = 64

2* = 2⁶

* = 6

Vậy * = 6.

\(3^x=9^{-6}.27^{-5}.81^8\)

\(3^x=\left(3^2\right)^{-6}.\left(3^3\right)^{-5}.\left(3^4\right)^8\)

\(3^x=3^{2.\left(-6\right)}.3^{3.\left(-5\right)}.3^{4.8}\)

\(3^x=3^{-12}.3^{-15}.3^{32}\)

\(3^x=3^{\left(-12\right)+\left(-15\right)+32}\)

\(3^x=3^5\)

\(\Rightarrow x=5\)

\(3^x=9^{-6}.27^{-5}.81^8\)

\(3^x=3^{-12}.3^{-15}.3^{32}=3^5\)

\(x=5\left(dox\ne\pm1;0\right)\)

Vậy...

M = \(\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+...+\frac{1}{6561}\)

=> 3M = \(1+\frac{1}{3}+\frac{1}{9}+...+\frac{1}{2187}\)

=> 3M - M = ( \(1+\frac{1}{3}+\frac{1}{9}+...+\frac{1}{2187}\)  ) - ( \(\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+...+\frac{1}{6561}\))

2M = 1 - \(\frac{1}{6561}\)

2M = \(\frac{6560}{6561}\)

=> M = \(\frac{3280}{6561}\)

\(M=\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\frac{1}{81}+.......+\frac{1}{6561}\)

\(\Rightarrow M=\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+\frac{1}{3^4}+.........+\frac{1}{3^8}\)

\(\Rightarrow3M=3\left(\frac{1}{3^2}+\frac{1}{3^3}+\frac{1}{3^4}+.........+\frac{1}{3^8}\right)\)

\(\Rightarrow3M=1+\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+............+\frac{1}{3^7}\)

\(\Rightarrow3M-M=1+\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+..........+\frac{1}{3^7}-\frac{1}{3}-\frac{1}{3^2}-\frac{1}{3^3}-.......-\frac{1}{3^8}\)

\(\Rightarrow2M=1-\frac{1}{3^8}\)

\(\Rightarrow M=\frac{1-\frac{1}{3^8}}{2}\)

Vậy M = \(\frac{1-\frac{1}{3^8}}{2}\)

\(M=\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\frac{1}{81}+...+\frac{1}{6561}\)

\(\Rightarrow3M=3\left(\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\frac{1}{81}+...+\frac{1}{6561}\right)\)

\(\Rightarrow3M=1+\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+...+\frac{1}{2187}\)

\(\Rightarrow3M-M=\left(1+\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+...+\frac{1}{2187}\right)-\left(\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\frac{1}{81}+\frac{1}{6561}\right)\)

\(\Rightarrow2M=1-\frac{1}{6561}\)

\(\Rightarrow2M=\frac{6560}{6561}\)

\(\Rightarrow M=\frac{3280}{6561}\)

3M=1+1/3+1/9+...+1/2187

2M=3M-M

2M=1-1/6561

2M=6560/6561

M=3280/6561

a/ \(27^x.9^x=9^{27}:81\)

\(\Leftrightarrow3^{3x}.3^{2x}=3^{54}:3^4\)

\(\Leftrightarrow3^{2x+3x}=3^{50}\)

\(\Leftrightarrow2x+3x=50\)

\(\Leftrightarrow5x=50\)

\(\Leftrightarrow x=10\)

Vậy ...

\(a.27^x.9^x=9^{27}:81\)

\(\left(3^3\right)^x.\left(3^2\right)^x=\left(3^2\right)^{27}:\left(3^2\right)^2\)

\(3^{3x}.3^{2x}=3^{50}\)

\(3^{3x+2x}=3^{50}\)

\(\Rightarrow3x+2x=50\)

\(x\left(3+2\right)=50\)

\(x=50:5=10\)

Vậy\(x=10\)

\(b.\left(\dfrac{12}{25}\right)^x=\left(\dfrac{5}{3}\right)^{-2}-\left(-\dfrac{3}{5}\right)^4\)

\(\left(\dfrac{12}{25}\right)^x=\dfrac{9}{25}-\dfrac{81}{625}\)

\(\left(\dfrac{12}{25}\right)^x=\dfrac{144}{625}\)( Đề sai )

\(M=\frac{3^6.3^8.5^4-3^{15}.5^{13-9}}{3^{12}.5^6+3^{18}.5^9}.2.\frac{25}{9}=\frac{3^{14}.5^4\left(1-3\right).2.5^2}{3^{12}.5^6\left(1+3^6.5^3\right).3^2}=\frac{-4}{1+3^6.5^3}\)

81^-2x.27^x= 9⁵

\(\left(3^4\right)^{-2x}.\left(3^3\right)^x=\left(3^2\right)^5\)

\(3^{-8x}.3^{3x}=3^{10}\)

\(3^{-8.x+3.x}=3^{10}\)

=> -8 .x +3.x = 10

x . [(-8) + 3] = 10

x . (-5) = 10

x = 10: (-5 )

x = -2

vậy x = (-2 )