1. Find all positive real numbers $x$ that satisfy the equation $x^{x \sqrt[3]{x}}=(x \sqrt[3]{x})^{x}$.
III/1. We take the logarithm of both sides of the equation and get \( x \sqrt[3]{x} \log x = x \log x^{\frac{4}{3}} = \frac{4}{3} x \log x \), which we rearrange to \( x \log x \left( \sqrt[3]{x} - \frac{4}{3} \right) = 0 \). Since \( x > 0 \), we get one solution of the equation from \( \log x = 0 \), which is \( x = 1 \), and the other from \( \sqrt[3]{x} = \frac{4}{3} \), which is \( x = \frac{64}{27} \).

Taking the logarithm and transforming the equation into the form of a product ........................................................................

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