What is root of equation 6*9^(1/x)-13*6^(1/x)+6*4^(1/x)=0?

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The root of the equation `6*9^(1/x)-13*6^(1/x)+6*4^(1/x)=0` has to be determined.

`6*9^(1/x)-13*6^(1/x)+6*4^(1/x)=0`

=> `6*3^(2/x)-13*3^(1/x)*2^(1/x)+6*2^(2/x)=0`

Let `3^(1/x) = X` and `2^(1/x) = Y`

=> `6*X^2 - 13*X*Y + 6*Y^2 = 0`

=> `(2*Y-3*X)*(3*Y-2*X) = 0`

2*Y-3*X = 0

=> `Y/X = 3/2`

=> `(2/3)^(1/x) = 3/2`

=> `1/x = -1`

=> x = -1

`3*Y-2*X = 0`

=> 3*Y = 2*X

=> `Y/X = 2/3`

=> `(2/3)^(1/x) = 2/3`

=> `1/x = 1`

=> x = 1

**The solution of the given equation is x = 1 and x = -1**

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