If ${ x }{ 1 }$ and ${ x }{ 2 }$ are the roots of equation ${ e }^{ 3/2 }.{ x }^{ 2\ell nx }={ x }^{ 4 }$, then the product of the roots of the equation is _______________.

Rewriting the equation e^(3/2) * x^(2 ln x) = x^4 using natural logarithms on both sides gives ln(e^(3/2)) + 2(ln x)^2 = 4 ln x. This rearranges to the quadratic equation in terms of ln x as 2(ln x)^2 - 4(ln x) + 3/2 = 0, or (ln x)^2 - 2(ln x) + 3/4 = 0. Using Vieta's formulas, the sum of the roots ln(x_1) + ln(x_2) is 2, which means ln(x_1 * x_2) = 2. Exponentiating both sides gives the product of the roots as e^2.

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