Solve for \( x \) in the equation:

\[ \left(\frac{5}{6}\right)^{4x} = \left(\frac{36}{25}\right)^{9-x} \]

Note: In the equation, \( 9-x \) is the exponent.

25 36 ​ = 5 2 6 2 ​ = ( 5 6 ​ ) 2 = ( 6 5 ​ ) − 2 t h ere f ore : ( 6 5 ​ ) 4 x = [ ( 6 5 ​ ) − 2 ] 9 − x ( 6 5 ​ ) 4 x = ( 6 5 ​ ) − 2 ( 9 − x ) ⟺ 4 x = − 2 ( 9 − x ) 4 x = − 2 ⋅ 9 − 2 ⋅ ( − x ) 4 x = − 18 + 2 x ∣ s u b t r a c t 2 x f ro m b o t h s i d es 2 x = − 18 ∣ d i v i d e b o t h s i d es b y 2 x = − 9 ​
U se : a − n = ( a 1 ​ ) n ( a n ) m = a n ⋅ m ( b a ​ ) n = b n a n ​

Answered by Anonymous | 2024-06-10

(25/36)^2x=(36/25)^9-x (36/25)^-2x=(36/25)^9-x
-2x=9-x -x=9 x=-9

Answered by ollieboyne | 2024-06-10

The solution to the equation ( 6 5 ​ ) 4 x = ( 25 36 ​ ) 9 − x is x = − 9 . By rewriting 25 36 ​ in terms of 6 5 ​ and equating the exponents, we can easily solve for x .
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Answered by Anonymous | 2024-12-26