To solve for x, we can first simplify the expression:

(3^x) * (2^(3/x)) = 24

Take the natural log of both sides to eliminate the exponents:

ln((3^x) * (2^(3/x))) = ln(24)

Using the properties of logarithms, this can be rewritten as:

ln(3^x) + ln(2^(3/x)) = ln(24)

Apply the power rule of logarithms:

x ln(3) + (3/x) ln(2) = ln(24)

Now we have a linear equation in terms of x. Rearrange the terms:

x ln(3) + 3 (ln(2) / x) = ln(24)

Multiply through by x to clear the fraction:

x^2 ln(3) + 3 ln(2) = x * ln(24)

Rearrange and factor:

x^2 ln(3) - x ln(24) + 3 * ln(2) = 0

This is a quadratic equation in terms of x, which can be solved using the quadratic formula:

x = (-(-ln(24)) ± √((-ln(24))^2 - 4 ln(3) 3 ln(2))) / (2 ln(3))

Calculating this expression will give you the values of x that satisfy the original equation.