[tex]4*3^{x} - 9*2^{x} = 5*6^{\frac{x}{2} }[/tex]

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To solve this equation, we first need to simplify it by writing 6 as 2 * 3.

So, the equation becomes:
[tex]43^{x} - 92^{x} = 5(23)^{\frac{x}{2}}[/tex]

Simplify further:
[tex]43^{x} - 92^{x} = 52^{\frac{x}{2}} 3^{\frac{x}{2}}[/tex]

Next, let's express everything in terms of a common base. We can see that the two bases are 2 and 3.

Therefore, rewrite 3^x as (3^2)^(x/2) and 2^x as (2^2)^(x/2) to have the same bases:

[tex]4(3^2)^{\frac{x}{2}} - 9(2^2)^{\frac{x}{2}} = 52^{\frac{x}{2}} 3^{\frac{x}{2}}[/tex]

Simplify further:
[tex]43^{x/2} - 92^{x/2} = 52^{x/2} 3^{x/2}[/tex]

Let's denote 3^(x/2) as a and 2^(x/2) as b.

The equation now becomes:
4a - 9b = 5ab.

Now we can solve for a and b with these simplified equations.